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A347135
a(n) = Sum_{d|n} A001615(n/d) * A069359(d).
5
0, 1, 1, 5, 1, 12, 1, 16, 7, 16, 1, 51, 1, 20, 18, 44, 1, 68, 1, 71, 22, 28, 1, 156, 11, 32, 33, 91, 1, 167, 1, 112, 30, 40, 26, 277, 1, 44, 34, 220, 1, 215, 1, 131, 110, 52, 1, 420, 15, 140, 42, 151, 1, 300, 34, 284, 46, 64, 1, 673, 1, 68, 138, 272, 38, 311, 1, 191, 54, 295, 1, 836, 1, 80, 162, 211, 38, 359, 1, 596
OFFSET
1,4
COMMENTS
Dirichlet convolution of A001615 (Dedekind psi function) with A069359.
Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A322577.
FORMULA
a(n) = Sum_{d|n} A001615(n/d) * A069359(d).
a(n) = Sum_{d|n} A001221(n/d) * A322577(d).
MATHEMATICA
Table[DivisorSum[n, PrimeNu[n/#]*Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, #/d] EulerPhi[d], {d, Divisors[#]}]&], {n, 80}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)
PROG
(PARI)
A001615(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347135(n) = sumdiv(n, d, A001615(n/d)*A069359(d));
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 23 2021
STATUS
approved