login
A349133
Dirichlet convolution of A003415 with A003958, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).
8
0, 1, 1, 5, 1, 8, 1, 17, 8, 12, 1, 32, 1, 16, 14, 49, 1, 43, 1, 52, 18, 24, 1, 100, 14, 28, 43, 72, 1, 87, 1, 129, 26, 36, 22, 151, 1, 40, 30, 168, 1, 119, 1, 112, 91, 48, 1, 276, 20, 103, 38, 132, 1, 194, 30, 236, 42, 60, 1, 323, 1, 64, 123, 321, 34, 183, 1, 172, 50, 183, 1, 443, 1, 76, 131, 192, 34, 215, 1, 472
OFFSET
1,4
LINKS
FORMULA
a(n) = Sum_{d|n} A003415(d) * A003958(n/d).
For all n >= 1, a(n) <= A349173(n).
MATHEMATICA
f1[p_, e_] := e/p; f2[p_, e_] := (p - 1)^e; a1[1] = 0; a1[n_] := n*Plus @@ (f1 @@@ FactorInteger[n]); a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, a1[#] * a2[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
PROG
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A349133(n) = sumdiv(n, d, A003415(d)*A003958(n/d));
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 09 2021
STATUS
approved