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

1, 2, 3, 4, 9, 10, 29, 30, 33, 34, 96, 98, 99, 154, 179, 180, 189, 216, 217, 242, 262, 294, 296, 428, 429, 446, 708, 756, 834, 1005, 1182, 1229, 1663, 1830, 1831, 1846, 1879, 2191, 2224, 2343, 2809, 3077, 3086, 3384, 3385, 3427, 3643, 3644, 3793, 3795, 4230

Offset: 1

Author

Gus Wiseman, Oct 07 2024

Keywords

• nonn

Comments

The prime numbers are (A000040):

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ...

with first differences (A001223):

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, ...

with second differences (A036263):

1, 0, 2, -2, 2, -2, 2, 2, -4, 4, -2, -2, 2, 2, 0, -4, 4, -2, -2, 4, -2, 2, 2, ...

with sorted first appearances at (A376656):

1, 2, 3, 4, 9, 10, 29, 30, 33, 34, 96, 98, 99, 154, 179, 180, 189, 216, 217, ...

Crossrefs

These are the sorted positions of first appearances in A036263.

For first differences we had A373400(n) + 1, except initial terms.

For prime-powers instead of prime numbers we have A376653/A376654.

For squarefree instead of prime numbers we have A376655, sorted firsts of A376590.

A000040 lists the prime numbers, differences A001223.

A005117 lists squarefree numbers, complement A013929 (differences A078147).

A333254 lists run-lengths of differences between consecutive primes.

For second differences: A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).

Cf. A064113, A175632, A251092, A258025, A258026, A333214, A373402, A376305, A376311.

Programs

Mathematica

q=Differences[Select[Range[1000],PrimeQ],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]