A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).

1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, 206, 207, 216, 244, 245, 304, 325, 327, 402, 466, 484, 605, 801, 880, 939, 1033, 1055, 1077, 1234, 1281, 1721, 1890, 1891, 1906, 1940, 1960, 1962, 2257, 2290, 2410, 2880, 3150

Offset: 1

Author

Gus Wiseman, Oct 06 2024

Keywords

• nonn

Examples

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with first appearances (A376653):
  1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, ...
		

Crossrefs

For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).

These are the sorted positions of first appearances in A376596.

The exclusive version is a(n) - 1 = A376654(n), except first term.

For squarefree instead of prime-power we have A376655.

A000961 lists prime-powers inclusive, exclusive A246655.

A001597 lists perfect-powers, complement A007916.

A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.

A064113 lists positions of adjacent equal prime gaps.

For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).

For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).

Cf. A053707, A069623, A093555, A174965, A251092, A333254, A361102.

Programs

Mathematica

q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]