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A376656 Sorted positions of first appearances in the second differences (A036263) of consecutive primes (A000040).

%I A376656 #9 Jun 26 2025 13:03:41
%S A376656 1,2,3,4,9,10,29,30,33,34,96,98,99,154,179,180,189,216,217,242,262,
%T A376656 294,296,428,429,446,708,756,834,1005,1182,1229,1663,1830,1831,1846,
%U A376656 1879,2191,2224,2343,2809,3077,3086,3384,3385,3427,3643,3644,3793,3795,4230
%N A376656 Sorted positions of first appearances in the second differences (A036263) of consecutive primes (A000040).
%C A376656 The prime numbers are (A000040):
%C A376656   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ...
%C A376656 with first differences (A001223):
%C A376656   1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, ...
%C A376656 with second differences (A036263):
%C A376656   1, 0, 2, -2, 2, -2, 2, 2, -4, 4, -2, -2, 2, 2, 0, -4, 4, -2, -2, 4, -2, 2, 2, ...
%C A376656 with sorted first appearances at (A376656):
%C A376656   1, 2, 3, 4, 9, 10, 29, 30, 33, 34, 96, 98, 99, 154, 179, 180, 189, 216, 217, ...
%t A376656 q=Differences[Select[Range[1000],PrimeQ],2];
%t A376656 Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
%Y A376656 These are the sorted positions of first appearances in A036263.
%Y A376656 For first differences we had A373400(n) + 1, except initial terms.
%Y A376656 For prime-powers instead of prime numbers we have A376653/A376654.
%Y A376656 For squarefree instead of prime numbers we have A376655, sorted firsts of A376590.
%Y A376656 A000040 lists the prime numbers, differences A001223.
%Y A376656 A005117 lists squarefree numbers, complement A013929 (differences A078147).
%Y A376656 A333254 lists run-lengths of differences between consecutive primes.
%Y A376656 For second differences: A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
%Y A376656 Cf. A064113, A175632, A251092, A258025, A258026, A333214, A373402, A376305, A376311.
%K A376656 nonn
%O A376656 1,2
%A A376656 _Gus Wiseman_, Oct 07 2024