A376656 - OEIS

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

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OFFSET

1,2

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 first 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, ...

LINKS

Table of n, a(n) for n=1..51.

MATHEMATICA

q=Differences[Select[Range[1000], PrimeQ], 2];

Select[Range[Length[q]], !MemberQ[Take[q, #-1], q[[#]]]&]

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.

Sequence in context: A291163 A180743 A162662 * A068334 A283874 A263092

Adjacent sequences: A376653 A376654 A376655 * A376657 A376658 A376659

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 07 2024

STATUS

approved