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        riddles >> general problem-solving / chatting / whatever >> 2017
        
(Message started by: towr on Dec 23^rd, 2016, 6:19am)

Title: 2017
Post by towr on Dec 23^rd, 2016, 6:19am

Does anyone have any interesting facts about the number 2017?

It's a prime and part of a sexy prime pair with 2011.
It's a zero of the Mertens function (https://en.wikipedia.org/wiki/Mertens_function)
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(2017) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(2017-1) + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(2017-2)

Title: Re: 2017
Post by rloginunix on Dec 23^rd, 2016, 5:35pm

Theme: 2+0+1=3 and 7:

333 + 3 + 333 + 3 + 333 + 7 + 333 + 3 + 333 + 3 + 333

Title: Re: 2017
Post by rloginunix on Dec 23^rd, 2016, 6:24pm

Also, 2017 is in the center of the immediate prime neighbours: 2011 on the left and 2027 on the right. Assembled together they all form yet another prime:
201120172027

Title: Re: 2017
Post by dudiobugtron on Dec 27^th, 2016, 11:14am

In hexadecimal it is 7e1, which in decimal scientific notation represents 7.

Title: Re: 2017
Post by towr on Dec 27^th, 2016, 1:09pm

Err, isn't that 70?

Title: Re: 2017
Post by dudiobugtron on Dec 27^th, 2016, 3:00pm

on 12/27/16 at 13:09:58, towr wrote:

Err, isn't that 70?

Er... yes. http://www.ocf.berkeley.edu/~wwu/YaBBImages/embarassed.gif

Title: Re: 2017
Post by rloginunix on Dec 28^th, 2016, 4:10pm

- for the record:
2017 = 9² + 44²
where nine is three squared and forty four is, of course, twenty two squared, ;D

- low intensity love affair with 9: recursively subtract the sum of digits of the current number from the current number:

2017 - 10 = 2007
and so, downward, we subtract: occasionally 27 but mostly 9 and 18 [at some point: 1818-18]

- interpreted as "a 24-hour period" it is 84 days and one hour:

2017 = 24x84 + 1 = 2016 + 1

- a busy beetle, running along a circumference, will count 5 full revolutions and then some:

2017 = 360x5 + 217
where the remainder corresponds to 7/6 past 7 on the face of an analogue clock

Any interest in constructing the first 100 natural numbers from the digits in 2017 and some basic operations?

0 = 2 x 0 x 1 x 7
1 = (2 + 0 - 1)⁷
2 = 2 + 0 x 1 x 7
3 = 2 + 1 + 0 x 7
4 = 7 - 1 - 0 - 2
5 = 7 - 2 - 0 x 1
6 = 7 - 1 - 2 x 0
7 = 7 + 2 x 0 x 1
8 = 7 + 1 + 2 x 0
9 = 7 + 2 + 0 x 1
10 = 2 + 0 + 1 + 7
11 = 2 + 0! + 1 + 7

Title: Re: 2017
Post by dudiobugtron on Dec 28^th, 2016, 5:07pm

12 = 12 + 0 x 7
;)
[hide]or 12 = 3 x 2 + 7 - 0![/hide]

on 12/28/16 at 16:10:35, rloginunix wrote:

- low intensity love affair with 9: recursively subtract the sum of digits of the current number from the current number:

2017 - 10 = 2007
and so, downward, we subtract: occasionally 27 but mostly 9 and 18 [at some point: 1818-18]

This is a result of the fact that the digits of multiples of 9 add to a multiple of 9. (and subtracting a multiple of 9 from another yields a multiple of 9 as well.)

I would be interested in whether there were any numbers (past a certain point) which didn't have a low-intensity love affair with 9.

Title: Re: 2017
Post by towr on Dec 28^th, 2016, 10:24pm

12 = 2*7-1-0!
13 = 2*7-1-0
14 = 2*7-1*0
15 = 2*7+1+0
16 = 2*7+1+0!
17 = 2*(7+1)+0!
18 = 2*(7+1+0!)

Title: Re: 2017
Post by rloginunix on Dec 29^th, 2016, 11:03am

on 12/28/16 at 17:07:46, dudiobugtron wrote:

[hide]or 12 = 3 x 2 + 7 - 0![/hide]

(sorry, can't use 3 explicitly (take towr's version or 12 = 7 + (2 + 1)! - 0!))

19 = C(7, 2) - 0! - 1 = 2 + 0 + 17 = 20 - 1⁷
20 = 7x(2 + 1) - 0!
21 = 7x(2 + 1) + 0

Separately, as a sum of consecutive primes (A000040 (http://oeis.org/A000040)) I only managed to assemble 2011:

2011 = 157 + ... + 211
Doesn't seem to work for 2017.

Title: Re: 2017
Post by rloginunix on Dec 29^th, 2016, 11:43am

With the help of a (C) program:

2011 = 661 + 673 + 677
2015 = 389 + 397 + 401 + 409 + 419
2016 = 71 + ... + 157
No cigar for 2017.

Title: Re: 2017
Post by dudiobugtron on Dec 29^th, 2016, 12:13pm

on 12/29/16 at 11:03:16, rloginunix wrote:

(sorry, can't use 3 explicitly (take towr's version or 12 = 7 + (2 + 1)! - 0!))

Oh gosh, I am obviously not on form in this thread.

Title: Re: 2017
Post by towr on Dec 29^th, 2016, 1:01pm

22 = 7x(2 + 1) + 0!
23 = (7-2-1)! - 0!
24 = (7-2-1)! + 0
25 = (7-2-1)! + 0!
26 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cgamma.gif(7-2) + 0! + 1
27 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif₁⁷x dx + 2 + 0!
28 = 7*2*(1+0!)
29 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif₁⁷x+http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cgamma.gif(2) dx - 0!
30 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif₁⁷x+http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cgamma.gif(2) dx + 0
31 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif₁⁷x+http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cgamma.gif(2) dx + 0!
32 = 2^(7-1-0!)
33 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gife^7/2 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif+ 0*1
34 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gife^7/2 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif+ 0 + 1
35 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gife^7/2 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif+ 0! + 1
36 = (7-1)² + 0
37 = (7-1)² + 0!

Title: Re: 2017
Post by rmsgrey on Dec 29^th, 2016, 4:50pm

on 12/28/16 at 17:07:46, dudiobugtron wrote:

I would be interested in whether there were any numbers (past a certain point) which didn't have a low-intensity love affair with 9.

It's not terribly interesting.

The sum of the digits of a number is the same value mod 9 as the number itself (because 10ⁿ*a_n = 1ⁿ*a_n = a_n (mod 9) ) so subtracting the sum of digits from the original number gives a multiple of 9 for any [e]positive[/e] integer

Title: Re: 2017
Post by rloginunix on Dec 29^th, 2016, 7:57pm

on 12/29/16 at 16:50:22, rmsgrey wrote:

It's not terribly interesting.

I concur, we made it too easy for ourselves.

Will the following "conservation of ordinal position" constraint make it more interesting:

- the digits in the construction must keep their yearly position

The construct with the smallest number of operat[ions]/[ors] shall win:

0 = 2 x 0 x 1 x 7
1 = (2 - 0 - 1)⁷
2 = 2 - 0 x 1 x 7
3 = -2 - 0! - 1 + 7
4 = -2 - 0 - 1 + 7
5 = -2 - 0 x 1 + 7
6 = -2 - 0 + 1 + 7
7 = 2 x 0 x 1 + 7
8 = 2 x 0 + 1 + 7
9 = 2 - 0 x 1 + 7
10 = 2 - 0 + 1 + 7
11 = 2 + 0! + 1 + 7
12 = 20 - 1 - 7
13 = (2 - 0 + 1)! + 7
14 = 20 + 1 - 7
15 = -2 + 0 + 17
16 = -2 + 0! + 17
17 = 2 x 0 + 17
18 = 2 - 0! + 17
19 = 2 + 0 + 17

Title: Re: 2017
Post by towr on Dec 29^th, 2016, 11:12pm

on 12/29/16 at 19:57:15, rloginunix wrote:

I concur, we made it too easy for ourselves.

He was talking about something else, though...

20 = 20 * 1⁷
21 = ~~(20+1)*7~~ 20 + 1⁷
22 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.giflog(20!) * 1 + log(7!)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif
23 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lceil.giflog(20!) * 1 + log(7!)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rceil.gif
24 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lceil.giflog(20!) + 1 + log(7!)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rceil.gif
25 = exp( (2+0) * ln(-1+ http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(7)) )
26 = 20 - 1 + 7
27 = 20 + 1 * 7
28 = 20 + 1 + 7
29 = 20 + 1 * http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lceil.gifln(7!)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rceil.gif
30 = 20 + 1 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lceil.gifln(7!)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rceil.gif
31 = 20 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lceil.gifln((1+7)!)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rceil.gif
32 = 2^{(-0! - 1 + 7)}
33 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gifexp(2^0-1*7)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif
34 = (2 + 0) * 17
35 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gifexp(2)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif* (-0!-1+7)
36 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gifexp(2) - 0!http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif* (-1+7)
37 = 20 + 17
38 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(201)) - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lceil.gifln(7)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rceil.gif
39 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(201)) - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gifln(7)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif
40 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(201)) * http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gifln(7)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif
41 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(201)) + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gifln(7)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif
42 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(201)) + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lceil.gifln(7)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rceil.gif
43 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(201)) + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.giflog(exp(7))http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif
44 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(201)) + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lceil.giflog(exp(7))http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rceil.gif
45 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gifexp(2)http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif* ln((0*1+7)!)

I'll leave the winning to someone else, it's hard enough getting this far.

Title: Re: 2017
Post by rloginunix on Dec 30^th, 2016, 3:10pm

Nice.

I am sure it's just a typo - for 21 you meant 20 + 1⁷.

on 12/29/16 at 23:12:02, towr wrote:

He was talking about something else, though...

Yeah, I know - programmer's habit to reuse things.

Speaking of reuse - let's reuse!

46 = -2 - 0 + (-1 + 7)!!
47 = -2 + 0! + (-1 + 7)!! = ((2 + 0!)!)!! - 1⁷
48 = (2 + 0!)! x (1 + 7)
49 = ((2 + 0!)! + 1) x 7
50 = 2 + 0 + (-1 + 7)!!
51 = (2 + 0!) x 17 = -20 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!) = !((2 + 0!)! - 1) + 7
52 = ~~2^{0! + 1}! + 7!!!~~ 2 x (0! + 1) x F₇
53 = ~~2^{h(0! + 1)} + 7!!!!~~ (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gife²http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif- 0!)!! - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/lfloor.gife¹http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rfloor.gif+ 7
54 = (2 + 0!)! + (-1 + 7)!!
55 = ((2 + 0 + 1)!)!! + 7
56 = ((2 + 0! + 1)!! x 7

Reference:
h(n) - hexagonal numbers A000384 (http://oeis.org/A000384)
n!! is A006882 (http://oeis.org/A006882)
n!!! is A007661 (http://oeis.org/A007661)
n!!!! is A007662 (http://oeis.org/A007662)
!n is A000166 (http://oeis.org/A000166)

[e]
see towr's comment below
[/e]

Title: Re: 2017
Post by towr on Dec 30^th, 2016, 11:56pm

Hmm, editing that mistake screwed up the whole post. The symbol script doesn't seem to work on this computer.

I also think that when you start needing references, it may be going a bit over the top. You can probably find some function f(x) that does exactly what you want somewhere on the web, or otherwise put it somewhere.
Maybe it's an idea to limit it to what wolframalpha will accept? That should be broad enough. So n!! is ok, but not n!!!

Title: Re: 2017
Post by rloginunix on Dec 31^st, 2016, 7:47pm

Deal!

I've checked - it computes subfactorials (!4 = 9, !5 = 44) and Fibonacci numbers as 'fibonacci 7' returning 13.

Before the New Year rolls through the East Coast (4!! = 8, 6!! = 48, 7!! = 105, http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(5041) = 71):

57 = -(((2 + 0 + 1)!)!! + 7!!
58 =
59 =
60 =
61 = -(!((2 + 0!)! - 1)) + 7!!
62 =
63 = !(2 + 0! + 1) x 7
64 = 2^{0 - 1 + 7}
65 = -(2 + 0!)! + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
66 =
67 =
68 = -2 - 0! + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
69 = -2 - 0 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
70 = -2 + 0! + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
71 = 2 x 0 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
72 = 2 - 0! + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
73 = 2 + 0 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
74 = 2 + 0! + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
75 =
76 =
77 = (2 + 0!)! + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(1 + 7!)
78 =
79 =
80 =
81 =
82 =
83 =
84 = -20 - 1 + 7!!
85 = -20/1 + 7!!
86 = -20 + 1 + 7!!
87 =
88 = 2 x !(-0! - 1 + 7)
89 =
90 =
91 =
92 =
93 =
94 = 2 x (-0! + (-1 + 7)!!)
95 =
96 = (2 + 0) x (-1 + 7)!!
97 = -((2^{0! + 1})!!) + 7!!
98 = -((2 + 0!)! + 1) + 7!!
99 = -((2 + 0!)! x 1) + 7!!
100 = -((2 + 0!)! - 1) + 7!!

Title: Re: 2017
Post by towr on Jan 2^nd, 2017, 12:13pm

57 = (2%)^(0 - 1) + 7
..
60 = ldexp(-2, 0!) + ldexp(1, totient(7))
..
62 = -2 + 0 + ldexp(1, totient(7))
..
66 = 2 + 0 + ldexp(1, totient(7))
67 = 2 + 0! + ldexp(1, totient(7))
..
80 = gcd(round(ldexp(2, 0!) / (1%))), 7!)
..
91 = -2 + 0! / (1%) - 7
92 = -2 + 0! / (1%) - totient(7)
93 = lb(2) * 0! / (1%) - 7
95 = 2 + 0! / 1% - 7
..
101 = lb(2) + 0! / ((1^7)%)
102 = 2 + 0! / ((1^7)%)
103 = ?
104 = -2 + ((0! / 1%) + totient(7))
105 = -2 + ((0! / 1%) + 7)
106 = lb(2) * (0! / (1%) + totient(7))
107 = lb(2) + (0! / (1%) + totient(7))
108 = 2 + (0! / (1%) + totient(7))
109 = 2 + (0! / (1%) + 7)

Title: Re: 2017
Post by dudiobugtron on Jan 2^nd, 2017, 12:16pm

...
2017 = 2017

;)

Title: Re: 2017
Post by rloginunix on Jan 2^nd, 2017, 7:07pm

103 = -2 - 0 x 1 + 7!!

I don't think I have the patience to slug it out to 2017!

(love % idea)

Title: Re: 2017
Post by towr on Jan 3^rd, 2017, 11:37am

58 = floor((20 + 1) * lb(7))
59 = ceil((20 + 1) * lb(7))

75 = floor(201 / sqrt(7))
76 = floor(sqrt(20) * 17)

78 = ceil(ldexp(20, 1) * ln(7))
79 = ceil((sqrt(20) + 1) / (7%))

81 = floor(20 * lb(17))
82 = floor(20 * sqrt(17))
83 = ceil(20 * sqrt(17))

87 = floor(log(20)^17)

89 = ceil(201^log(7))
90 = (totient(20)^-1) * gamma(7)
94 = ((2^0) / 1%) - totient(7)

103 = floor(201 / ln(7))
110 = ceil(lb(201) / (7%))
111 = floor(totient(201) * log(7))
112 = ldexp(totient(20), 1) * 7
113 = ceil(ldexp(20, 1) * lb(7))
114 = (20 - 1) * totient(7)
115 = 20! mod (-1 + gamma(7))
116 = floor(ln(20) * exp(1) / (7%))
117 = floor(20 / (17%))
118 = ceil(20 / (17%))
119 = floor((20 - (1%)) * totient(7))
120 = 20 * (1 * totient(7))
121 = ceil((20 - exp(1)) * 7)
122 = ceil((2^((0! - 1%) * 7)))
123 = floor((2 - (0 + 1%))^7)
124 = ldexp(-2, 0!) + ldexp(1, 7)
125 = totient(201) - 7
126 = (20 + 1) * totient(7)
127 = !20 mod (1 + gamma(7))
128 = ldexp(gcd(20, 1), 7)
129 = (2^0) + ldexp(1, 7)
130 = 2 + 0 + ldexp(1, 7)
131 = 2 + 0! + ldexp(1, 7)
132 = totient(201) mod floor(exp(7))
133 = (20 - 1) * 7
134 = floor((totient(201) + lb(7)))
135 = ceil((totient(201) + lb(7)))
136 = totient(20) * 17
137 = ceil(((20 + exp(1)) * totient(7)))
138 = totient(201) + totient(7)
139 = totient(201) + 7
140 = 20 * 1 * 7
141 = ceil((20 * (1% + 7)))
142 = ceil((log(2)^(0 - sqrt(17))))
143 = ceil((ldexp(20, -1) / 7%))
144 = 20% / 1 / gamma(7)
145 = ceil(20% * (1 + gamma(7)))
146 = 20 / (1%) - !7
147 = (20 + 1) * 7
148 = (20 + ldexp(1, 7))
149 = floor(log(201)^totient(7))
150 = (ldexp(2%, 0!)^-1) * totient(7)
151 = ceil(ln(20) * (1%) * 7!)
152 = floor(20 * exp(1) * lb(7))
153 = floor(ldexp(20% + 1, 7))
154 = ceil(ldexp(20% + 1, 7))
155 = floor((sqrt(20)^exp(1)) * sqrt(7))
156 = floor(totient(201) / log(7))
157 = ceil(totient(201) / log(7))
158 = floor((ldexp(2, 0!) / 1%)^log(7))
159 = floor((20 + exp(1)) * 7)
160 = 20 * (1 + 7)
161 = ceil(sqrt(20)^-1 * gamma(7))
162 = (gamma(20) * (1%)) mod !7
163 = floor(lb(20) / (1% * sqrt(7)))
164 = ceil(lb(20) / (1% * sqrt(7)))
165 = floor(ldexp((log(20) - 1%), 7))
166 = floor(ldexp(log(20), 1 * 7))
167 = ceil(ldexp(log(20), 1 * 7))
168 = 20! mod (1 + gamma(7))
169 = floor(201 * log(7))
170 = gamma(20) mod (-1 + !7)
171 = ceil(log(2)^(0 + exp(1) - 7))
172 = floor((totient(20) - 1)^sqrt(7))
173 = ceil((totient(20) - 1)^sqrt(7))
174 = floor(sqrt(201)^ln(7))
175 = (ldexp(2%, 0!)^-1) * 7
176 = ceil(2 + 0 + ldexp(exp(1), totient(7)))
177 = floor(ldexp(ln(2), (0 + 1 + 7)))
178 = ceil(ldexp(ln(2), (0 + 1 + 7)))
179 = floor(ldexp(sqrt(2) - 0 - (1%), 7))
180 = (ldexp(2, 0!)^-1) * gamma(7)
181 = floor(ldexp(sqrt(2), 0 + 1 * 7))
182 = ceil(ldexp(sqrt(2), 0 + 1 * 7))
183 = ceil(ldexp(sqrt(2) + 0 + (1%), 7))
184 = floor(ldexp(ln(2)^(0 - 1), 7))
185 = ceil(ldexp(ln(2)^(0 - 1), 7))
186 = 2 * (0! / (1%) - 7)
187 = ceil((ldexp(ln(2), 0!)^totient(17)))
188 = 2 * (0! / (1%) - totient(7))
189 = floor(ldexp(gamma(20)^(1%), 7))
190 = ceil(ldexp(gamma(20)^(1%), 7))
191 = floor(ldexp(ln(20), 1 * totient(7)))
192 = ldexp(2 + 0 + 1, totient(7))
193 = 2 / (0 + (1%)) - 7
194 = 201 - 7
195 = 201 - totient(7)
196 = floor((20 / exp(1))^sqrt(7))
197 = ceil((20 / exp(1))^sqrt(7))
198 = floor(201 - lb(7))
199 = ceil((201 - lb(7)))
200 = 2 / (0 + (1^7)%)

(I haven't thrown them all through wolfram-alpha yet. it sometimes reacts a bit different than python)

Title: Re: 2017
Post by towr on Jan 3^rd, 2017, 12:02pm

in general

2k+0 = lb(2)+log(0!/(1%....%))+7
2k+1 = 2+log(0!/(1%....%))+7

(Which is definitely not the least number of operators for most)

Title: Re: 2017
Post by rloginunix on Jan 3^rd, 2017, 1:30pm

I've checked till 120 - wolframalpha has issues with 87, 89, 94 and 111.

With 94 it's just a typo - extra parens are needed around 1%: "((2^0) / (1%)) - totient(7)"

Title: Re: 2017
Post by towr on Jan 3^rd, 2017, 10:20pm

Oh yeah, wolfram alpha interprets log() as ln(), but it does have log10().
I suppose I could try to find ones without log. One the one hand, it does supply the function, but on the other hand only in a form that contains digits.

And in many cases I need to put parenthesis around percentages, so 94 = (2^0) / (1%) - totient(7) should work.

Title: Re: 2017
Post by rloginunix on Jan 4^th, 2017, 7:05pm

Aah, an international pool of (interdisciplinary) confusion - logarithmic notation ...

Separately, not hugely interesting, just some trivia I thought of while driving: if we construct a sum of powers of two then we get a nicely ordered set of initial odd integers:
2² + 2⁰ + 2¹ + 2⁷ = 135

The sum of (decimal) digits in bases 2 and 6 are equal (to 7), ditto bases 3 and 5 (9):

(2017)₂ = 11111100001, sums to 7
(2017)₆ = 13201, sums to 7
(2017)₃ = 2202201, sums to 9
(2017)₅ = 31032, sums to 9

Title: Re: 2017
Post by towr on Jan 4^th, 2017, 10:35pm

In bases 10, 19, 22, 25, 29, 33, 37, 43, 49, 57, 64, 73, 85, 97, 113, 127, 145, 169, 225, 253, 289, 337, 505, 673, 1009 the digits sum to 10 (in the same base)
e.g 2017₁₀ = [1][1008]₁₀₀₉ => 1₁₀₀₉+1008₁₀₀₉=10₁₀₀₉ = 1009₁₀