A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

1, 4, 9, 12, 18, 24, 34, 47, 60, 79, 117, 178, 198, 206, 215, 244, 311, 402, 465, 614, 782, 1078, 1109, 1234, 1890, 1939, 1961, 2256, 2290, 3149, 3377, 3460, 3502, 3722, 3871, 4604, 4694, 6634, 8073, 8131, 8793, 12370, 12661, 14482, 14990, 15912, 17140, 19166

Offset: 1

Author

Gus Wiseman, Sep 22 2024

Keywords

• nonn

Examples

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    34: {1,7}
    47: {15}
    60: {1,1,2,3}
    79: {22}
   117: {2,2,6}
   178: {1,24}
   198: {1,2,2,5}
   206: {1,27}
   215: {3,14}
   244: {1,1,18}
		

Crossrefs

For compression instead of sorted firsts we have A376308.

For run-lengths instead of sorted firsts we have A376309.

For run-sums instead of sorted firsts we have A376310.

The version for squarefree numbers is the unsorted version of A376311.

The unsorted version is A376341.

A000040 lists the prime numbers, differences A001223.

A000961 and A246655 list prime-powers, first differences A057820.

A003242 counts compressed compositions, ranks A333489.

A005117 lists squarefree numbers, differences A076259.

A024619 and A361102 list non-prime-powers, first differences A375708.

A116861 counts partitions by compressed sum, by compressed length A116608.

Cf. A001597, A006549, A007916, A025475, A037201, A053289, A078147, A110969, A120430, A174965, A373948, A375706.

Programs

Mathematica

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