A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81

Offset: 1

Author

Gus Wiseman, Sep 28 2024

Keywords

• nonn

Examples

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with positions of first appearances (A376268):
  1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, ...
		

Crossrefs

These are the sorted positions of first appearances in A053289 (union A023055).

The complement is A376519.

A053707 lists first differences of consecutive prime-powers.

A333254 lists run-lengths of differences between consecutive primes.

Other families of numbers and their first differences:

For prime numbers (A000040) we have A001223.

For composite numbers (A002808) we have A073783.

For nonprime numbers (A018252) we have A065310.

For perfect powers (A001597) we have A053289.

For non-perfect-powers (A007916) we have A375706.

For squarefree numbers (A005117) we have A076259.

For nonsquarefree numbers (A013929) we have A078147.

For prime-powers inclusive (A000961) we have A057820.

For prime-powers exclusive (A246655) we have A057820(>1).

For non-prime-powers inclusive (A024619) we have A375735.

For non-prime-powers exclusive (A361102) we have A375708.

Cf. A025475, A045542, A052410, A069623, A093555, A174965, A216765, A303707, A305630, A305631, A375736.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
q=Differences[Select[Range[1000],perpowQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]