A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

4, 9, 8, 25, 16, 49, 32, 81, 64, 121, 128, 169, 256, 625, 512, 289, 1024, 361, 2048, 1444, 1331, 529, 5324, 2116, 2197, 4232, 8788, 841, 17576, 961, 7569, 3844, 4913, 7688, 19652, 1369, 6859, 5476, 12321, 1681, 34225, 1849, 15129, 7396, 12167, 2209, 46225, 8836, 19881

Offset: 2

Author

Michel Marcus, Dec 31 2016

Keywords

• nonn

Links

• Michel Marcus, Table of n, a(n) for n = 2..500

Crossrefs

Cf. A001414 (sopfr), A008472 (sopf), A001248, A280163.

A multiplicative version is A064549 (sorted A001694), firsts of A003557.

For length instead of sum we have A151821.

These are the positions of first appearances in A280292 = A001414 - A008472.

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

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

For prime exponents instead of factors we have A380989, firsts of A380958.

The sorted version is A381075.

For product instead of sum see A381076, sorted firsts of A066503.

A000040 lists the primes, differences A001223.

A005117 lists squarefree numbers, complement A013929.

A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).

A056239 adds up prime indices, row sums of A112798.

Cf. A000720, A001222, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A364916, A366528.

Programs

Mathematica

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,1000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[q,k][[1,1]],{k,2,mnrm[q/.(0->1)]}] (* Gus Wiseman, Feb 20 2025 *)

PARI

sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]*f[j,2]);
sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]);
a(n) = {my(k = 2); while (sopfr(k) - sopf(k) != n, k++); k;}

Formula

For p prime, a(p) = p^2 (see A001248).