A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

1, 4, 8, 16, 27, 64, 81, 256, 243, 529, 729, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 0

Author

Gus Wiseman, Feb 12 2025

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also the position of first appearance of n in A374248.

Examples

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   243: {2,2,2,2,2}
   529: {9,9}
   729: {2,2,2,2,2,2}
   961: {11,11}
  1369: {12,12}
  1681: {13,13}
  1849: {14,14}
  2209: {15,15}
		

Crossrefs

For length instead of sum we have A151821.

For factors instead of indices we have A280286 (sorted A381075), firsts of A280292.

Counting partitions by this statistic gives A364916.

Positions of first appearances in A380955.

The sorted version is A380957.

For product instead of sum we have firsts of A380986.

A multiplicative version is A380987 (sorted A380988), firsts of A290106.

For prime multiplicities instead of prime indices we have A380989, firsts of A380958.

A000040 lists the primes, differences A001223.

A005117 lists squarefree numbers, complement A013929.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, length A001222.

A304038 lists distinct prime indices, sum A066328, length A001221.

Cf. A000720, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A325033, A366528, A366749, A374248.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Total[prix[n]]-Total[Union[prix[n]]],{n,1000}];
Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]

Formula

After a(12) = 961, this appears to converge to prime(n)^2.