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towr
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 2017   « on: Dec 23rd, 2016, 6:19am » Quote Modify

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
(2017) = (2017-1) + (2017-2)
 « Last Edit: Dec 23rd, 2016, 6:28am by towr » IP Logged

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 Re: 2017   « Reply #1 on: Dec 23rd, 2016, 5:35pm » Quote Modify

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

333 + 3 + 333 + 3 + 333 + 7 + 333 + 3 + 333 + 3 + 333
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 Re: 2017   « Reply #2 on: Dec 23rd, 2016, 6:24pm » Quote Modify

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
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dudiobugtron
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 Re: 2017   « Reply #3 on: Dec 27th, 2016, 11:14am » Quote Modify

In hexadecimal it is 7e1, which in decimal scientific notation represents 7.
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towr
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 Re: 2017   « Reply #4 on: Dec 27th, 2016, 1:09pm » Quote Modify

Err, isn't that 70?
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dudiobugtron
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 Re: 2017   « Reply #5 on: Dec 27th, 2016, 3:00pm » Quote Modify

on Dec 27th, 2016, 1:09pm, towr wrote:
 Err, isn't that 70?

Er... yes.
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 Re: 2017   « Reply #6 on: Dec 28th, 2016, 4:10pm » Quote Modify

- for the record:
2017 = 92 + 442

where nine is three squared and forty four is, of course, twenty two squared,

- 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
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dudiobugtron
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 Re: 2017   « Reply #7 on: Dec 28th, 2016, 5:07pm » Quote Modify

12 = 12 + 0 x 7

or 12 = 3 x 2 + 7 - 0!

on Dec 28th, 2016, 4:10pm, 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.
 « Last Edit: Dec 28th, 2016, 5:07pm by dudiobugtron » IP Logged
towr
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 Re: 2017   « Reply #8 on: Dec 28th, 2016, 10:24pm » Quote Modify

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!)
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 Re: 2017   « Reply #9 on: Dec 29th, 2016, 11:03am » Quote Modify

on Dec 28th, 2016, 5:07pm, dudiobugtron wrote:
 or 12 = 3 x 2 + 7 - 0!

(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) I only managed to assemble 2011:

2011 = 157 + ... + 211

Doesn't seem to work for 2017.
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 Re: 2017   « Reply #10 on: Dec 29th, 2016, 11:43am » Quote Modify

With the help of a (C) program:

2011 = 661 + 673 + 677

2015 = 389 + 397 + 401 + 409 + 419

2016 = 71 + ... + 157

No cigar for 2017.
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dudiobugtron
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 Re: 2017   « Reply #11 on: Dec 29th, 2016, 12:13pm » Quote Modify

on Dec 29th, 2016, 11:03am, 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.
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towr
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 Re: 2017   « Reply #12 on: Dec 29th, 2016, 1:01pm » Quote Modify

22 = 7x(2 + 1) + 0!
23 = (7-2-1)! - 0!
24 = (7-2-1)! + 0
25 = (7-2-1)! + 0!
26 = (7-2) + 0! + 1
27 = 17x dx + 2 + 0!
28 = 7*2*(1+0!)
29 = 17x+(2) dx - 0!
30 = 17x+(2) dx + 0
31 = 17x+(2) dx + 0!
32 = 2(7-1-0!)
33 = e7/2 + 0*1
34 = e7/2 + 0 + 1
35 = e7/2 + 0! + 1
36 = (7-1)2 + 0
37 = (7-1)2 + 0!
 « Last Edit: Dec 29th, 2016, 1:27pm by towr » IP Logged

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rmsgrey
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 Re: 2017   « Reply #13 on: Dec 29th, 2016, 4:50pm » Quote Modify

on Dec 28th, 2016, 5:07pm, 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*an = 1n*an = an (mod 9) ) so subtracting the sum of digits from the original number gives a multiple of 9 for any [e]positive[/e] integer
 « Last Edit: Dec 30th, 2016, 5:40am by rmsgrey » IP Logged
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 Re: 2017   « Reply #14 on: Dec 29th, 2016, 7:57pm » Quote Modify

on Dec 29th, 2016, 4:50pm, 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
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towr
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 Re: 2017   « Reply #15 on: Dec 29th, 2016, 11:12pm » Quote Modify

on Dec 29th, 2016, 7:57pm, 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 = log(20!) * 1 + log(7!)
23 = log(20!) * 1 + log(7!)
24 = log(20!) + 1 + log(7!)
25 = exp( (2+0) * ln(-1+ (7)) )
26 = 20 - 1 + 7
27 = 20 + 1 * 7
28 = 20 + 1 + 7
29 = 20 + 1 * ln(7!)
30 = 20 + 1 + ln(7!)
31 = 20 + ln((1+7)!)
32 = 2(-0! - 1 + 7)
33 = exp(20-1*7)
34 = (2 + 0) * 17
35 = exp(2)* (-0!-1+7)
36 = exp(2) - 0!* (-1+7)
37 = 20 + 17
38 = ((201)) - ln(7)
39 = ((201)) - ln(7)
40 = ((201)) * ln(7)
41 = ((201)) + ln(7)
42 = ((201)) + ln(7)
43 = ((201)) + log(exp(7))
44 = ((201)) + log(exp(7))
45 = exp(2)* ln((0*1+7)!)

I'll leave the winning to someone else, it's hard enough getting this far.
 « Last Edit: Jan 1st, 2017, 11:01am by towr » IP Logged

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 Re: 2017   « Reply #16 on: Dec 30th, 2016, 3:10pm » Quote Modify

Nice.

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

on Dec 29th, 2016, 11:12pm, 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 + (1 + 7!) = !((2 + 0!)! - 1) + 7
52 = 20! + 1! + 7!!! 2 x (0! + 1) x F7
53 = 2h(0! + 1) + 7!!!! (e2- 0!)!! - e1+ 7
54 = (2 + 0!)! + (-1 + 7)!!
55 = ((2 + 0 + 1)!)!! + 7
56 = ((2 + 0! + 1)!! x 7

Reference:
h(n) - hexagonal numbers A000384
n!! is A006882
n!!! is A007661
n!!!! is A007662
!n is A000166

[e]
see towr's comment below
[/e]
 « Last Edit: Dec 31st, 2016, 7:02pm by rloginunix » IP Logged
towr
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 Re: 2017   « Reply #17 on: Dec 30th, 2016, 11:56pm » Quote Modify

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!!!

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 Re: 2017   « Reply #18 on: Dec 31st, 2016, 7:47pm » Quote Modify

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, (1 + 7!) = (5041) = 71):

57 = -(((2 + 0 + 1)!)!! + 7!!
58 =
59 =
60 =
61 = -(!((2 + 0!)! - 1)) + 7!!
62 =
63 = !(2 + 0! + 1) x 7
64 = 20 - 1 + 7
65 = -(2 + 0!)! + (1 + 7!)
66 =
67 =
68 = -2 - 0! + (1 + 7!)
69 = -2 - 0 + (1 + 7!)
70 = -2 + 0! + (1 + 7!)
71 = 2 x 0 + (1 + 7!)
72 = 2 - 0! + (1 + 7!)
73 = 2 + 0 + (1 + 7!)
74 = 2 + 0! + (1 + 7!)
75 =
76 =
77 = (2 + 0!)! + (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 = -((20! + 1)!!) + 7!!
98 = -((2 + 0!)! + 1) + 7!!
99 = -((2 + 0!)! x 1) + 7!!
100 = -((2 + 0!)! - 1) + 7!!
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towr
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 Re: 2017   « Reply #19 on: Jan 2nd, 2017, 12:13pm » Quote Modify

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)
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dudiobugtron
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 Re: 2017   « Reply #20 on: Jan 2nd, 2017, 12:16pm » Quote Modify

...
2017 = 2017

 « Last Edit: Jan 2nd, 2017, 12:16pm by dudiobugtron » IP Logged
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 Re: 2017   « Reply #21 on: Jan 2nd, 2017, 7:07pm » Quote Modify

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

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

(love % idea)
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 Re: 2017   « Reply #22 on: Jan 3rd, 2017, 11:37am » Quote Modify

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)
 « Last Edit: Jan 3rd, 2017, 11:58am by towr » IP Logged

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towr
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 Re: 2017   « Reply #23 on: Jan 3rd, 2017, 12:02pm » Quote Modify

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)
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 Re: 2017   « Reply #24 on: Jan 3rd, 2017, 1:30pm » Quote Modify

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)"
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