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A349132
a(n) = Sum_{d|n} psi(d) * A003958(n/d), where A003958 is fully multiplicative with a(p) = (p-1), and psi is Dedekind psi function, A001615.
7
1, 4, 6, 10, 10, 24, 14, 22, 24, 40, 22, 60, 26, 56, 60, 46, 34, 96, 38, 100, 84, 88, 46, 132, 70, 104, 84, 140, 58, 240, 62, 94, 132, 136, 140, 240, 74, 152, 156, 220, 82, 336, 86, 220, 240, 184, 94, 276, 140, 280, 204, 260, 106, 336, 220, 308, 228, 232, 118, 600, 122, 248, 336, 190, 260, 528, 134, 340, 276, 560
OFFSET
1,2
COMMENTS
Dirichlet convolution of A003958 with Dedekind psi function, A001615.
LINKS
FORMULA
a(n) = Sum_{d|n} A001615(d) * A003958(n/d).
a(n) = A327251(n) - A348982(n).
For all n >= 1, a(n) <= A349172(n).
Multiplicative with a(p^e) = (p+1)*p^e - p*(p-1)^e. - Amiram Eldar, Nov 09 2021
MATHEMATICA
f[p_, e_] := (p + 1)*p^e - p*(p - 1)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 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)));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349132(n) = sumdiv(n, d, A001615(d)*A003958(n/d));
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Nov 09 2021
STATUS
approved