A295746 Distinct second differences in the sequence of primes in order of appearance.
1, 0, 2, -2, -4, 4, 10, -10, 8, -8, 6, 16, -12, -16, 12, -14, -20, 28, -28, -6, 22, 14, 20, 26, -26, -22, -24, 24, -18, 18, 32, -32, -30, 40, -40, 30, -38, -34, 46, 38, 50, -46, 34, 68, -64, -44, 58, -58, -48, -50, 48, 52, -54, 44, -52, 36, -42, 56
Offset: 1
Keywords
Examples
From _Jon E. Schoenfield_, Jan 15 2018: (Start)
The first several primes and their 1st and 2nd differences are as follows:
.
k prime(k) 1st difference 2nd difference
-- -------- -------------- -------------------
1 2
3 - 2 = 1
2 3 2 - 1 = 1 (new)
5 - 3 = 2
3 5 2 - 2 = 0 (new)
7 - 5 = 2
4 7 4 - 2 = 2 (new)
11 - 7 = 4
5 11 2 - 4 = -2 (new)
13 - 11 = 2
6 13 4 - 2 = 2 (repeat)
17 - 13 = 4
7 17 2 - 4 = -2 (repeat)
19 - 17 = 2
8 19 4 - 2 = 2 (repeat)
23 - 19 = 4
9 23 6 - 4 = 2 (repeat)
29 - 23 = 6
10 29 2 - 6 = -4 (new)
31 - 29 = 2
11 31
.
and the 2nd differences that are not repeats of 2nd differences encountered earlier are, in order of appearance, 1, 0, 2, -2, -4, ..., i.e., the terms of this sequence. (End)
Links
- Robert G. Wilson v, Table of n, a(n) for n = 1..358 (first 97 terms from Edward Bernstein)
Programs
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Maple
P:= select(isprime, [2,seq(i,i=3..10^6,2)]): DP:= P[2..-1]-P[1..-2]: DDP:= DP[2..-1]-DP[1..-2]: ListTools:-MakeUnique(DDP); # Robert Israel, Jan 15 2018
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Mathematica
DeleteDuplicates@ Differences[Prime@ Range[10^4], 2] (* Michael De Vlieger, Dec 09 2017 *)
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PARI
first(n) = { my(res = vector(n), i=3, j=3); res[1]=1; res[2]=0; while(i<=n, my(d=prime(j+2)+prime(j)-2*prime(j+1)); if(!setsearch(Set(res), d), res[i]=d; i++); j++); res; } \\ Iain Fox, Nov 30 2017
Comments