A381075 Sorted positions of first appearances in A280292 (sum of prime factors minus sum of distinct prime factors).

1, 4, 8, 9, 16, 25, 32, 49, 64, 81, 121, 128, 169, 256, 289, 361, 512, 529, 625, 841, 961, 1024, 1331, 1369, 1444, 1681, 1849, 2048, 2116, 2197, 2209, 2809, 3481, 3721, 3844, 4232, 4489, 4913, 5041, 5324, 5329, 5476, 6241, 6859, 6889, 7396, 7569, 7688, 7921

Offset: 1

Author

Gus Wiseman, Feb 18 2025

Keywords

• nonn

Examples

			The initial terms of A280292 are (0,0,0,2,0,0,0,4,3,0,0,2,0,0,0,6,0,3,0,2,0,0,0,4,5,0,6,2,...), wherein a value appears for the first time at positions 1, 4, 8, 9, 16, 25, ...
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..391

Crossrefs

For length instead of sum we have A151821.

The unsorted version is A280286, firsts of A280292.

For indices instead of factors we have A380957 (unsorted A380956), firsts of A380955.

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

For prime multiplicities instead of factors see A380989, firsts of A380958.

For product instead of sum we have A381076, sorted firsts of A066503.

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, counted by A001222.

A364916 counts partitions by (sum minus sum of distinct parts).

Cf. A000720, A001414, A008472, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A366528.

Programs

Mathematica

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

PARI

f(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2] - f[j, 1]); \\ A280292
lista(nn) = my(v=Set(vector(nn, i, f(i))), list=List()); for (i=1, #v, my(k=1); while(f(k) != v[i], k++); listput(list, k)); vecsort(Vec(list)); \\ Michel Marcus, Apr 15 2025

Formula

Sorted positions of first appearances in A001414 - A008472.