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A348983
a(n) = Sum_{d|n} A038040(d) * A322582(n/d), where A038040(n) = n*d(n), A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).
7
0, 1, 1, 7, 1, 14, 1, 31, 11, 20, 1, 80, 1, 26, 23, 111, 1, 109, 1, 122, 29, 38, 1, 328, 19, 44, 76, 164, 1, 250, 1, 351, 41, 56, 35, 565, 1, 62, 47, 514, 1, 334, 1, 248, 208, 74, 1, 1128, 27, 245, 59, 290, 1, 650, 47, 700, 65, 92, 1, 1336, 1, 98, 274, 1023, 53, 502, 1, 374, 77, 490, 1, 2213, 1, 116, 302, 416, 53
OFFSET
1,4
COMMENTS
Dirichlet convolution of A322582 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A348980.
Dirichlet convolution of sigma (A000203) with A348981.
LINKS
FORMULA
a(n) = Sum_{d|n} A038040(n/d) * A322582(d).
a(n) = Sum_{d|n} d * A348980(n/d).
a(n) = Sum_{d|n} A000203(d) * A348981(n/d).
For all n >= 1, a(n) <= A349123(n) <= A349143(n).
MATHEMATICA
f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#])*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
PROG
(PARI)
A038040(n) = (n*numdiv(n));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348983(n) = sumdiv(n, d, A038040(n/d)*A322582(d));
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 08 2021
STATUS
approved