A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

1, 2, 3, 5, 6, 10, 17, 43, 70, 1077, 6635, 12369, 43578, 105102700

Offset: 1

Author

Gus Wiseman, Sep 04 2024

Keywords

• nonn
• more

Comments

The corresponding prime-powers A246655(a(n)) are given by A006549.

From A006549, it is not known whether this sequence is infinite.

Examples

			The fifth prime-power is 7 and the sixth is 8, so 5 is in the sequence.
		

Crossrefs

For nonprime numbers (A002808) we have A375926, differences A373403.

Positions of 1's in A057820.

First differences are A373671.

For nonsquarefree numbers we have A375709, differences A373409.

For non-prime-powers we have A375713.

For non-perfect-powers we have A375740.

For squarefree numbers we have A375927, differences A373127.

Prime-powers:

- terms: A000961, complement A024619.

- differences: A057820.

- runs: A373675, A373673, A373674, A174965

- anti-runs: A373576, A120430, A006549, A373671

Non-prime-powers:

- terms: A361102

- differences: A375708

- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670).

- anti-runs: A373679, A373575, A255346, A373672

A000040 lists all of the primes, differences A001223.

A025528 counts prime-powers up to n.

Cf. A007916, A014963, A027833, A046933, A053289, A093555, A375714.

Programs

Mathematica

Join@@Position[Differences[Select[Range[100],PrimePowerQ]],1]

Formula

Numbers k such that A246655(k+1) - A246655(k) = 1.

The inclusive version is a(n) + 1 shifted.

Extensions

a(14) from Amiram Eldar, Sep 24 2024