A335079 Row sums of A335078.

1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 3, 4, 1, 6, 1, 7, 2, 2, 2, 11, 1, 2, 2, 8, 1, 6, 1, 4, 4, 2, 1, 16, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 16, 1, 2, 4, 13, 2, 6, 1, 4, 2, 6, 1, 24, 1, 2, 4, 4, 2, 6, 1, 16, 5, 2, 1, 16, 2, 2, 2, 8, 1, 16

Offset: 2

Author

Stefano Spezia, May 23 2020

Keywords

• nonn

Links

• Stefano Spezia, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001222 (Omega function), A334996, A334997, A335078.

Programs

Mathematica

tau[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; (* A334997 *)
Nd[n_, m_]:=Sum[(-1)^k*Binomial[m, k]*tau[n, m-k-1], {k, 0, m-1}]; (* A334996 *)
T[n_,k_]:=1/k*DivisorSum[k,EulerPhi[#]*Nd[n^(1/#),k/#]&,IntegerQ[n^(1/#)]&]; (* A335078 *)
Table[Sum[T[n,m],{m,1,PrimeOmega[n]}],{n,2,90}]

PARI

TT(n, k) = if (k==0, 1, sumdiv(n, d, TT(d, k-1))); \\ A334996
U(n, m) = sum(k=0, m-1, (-1)^k*binomial(m, k)*TT(n, m-k-1));
T(n, k) = my(p); (1/k)*sumdiv(k, d, if (ispower(n, d, &p), eulerphi(d)*U(p, k/d)));
row(n) = vector(bigomega(n), k, T(n,k)); \\ A335078
a(n) = vecsum(row(n)); \\ Michel Marcus, May 25 2020