A347125 Möbius transform of A341529, sigma(n) * A003961(n).

1, 8, 19, 54, 41, 152, 87, 342, 305, 328, 155, 1026, 237, 696, 779, 2106, 341, 2440, 459, 2214, 1653, 1240, 695, 6498, 1477, 1896, 4675, 4698, 929, 6232, 1183, 12798, 2945, 2728, 3567, 16470, 1557, 3672, 4503, 14022, 1805, 13224, 2067, 8370, 12505, 5560, 2543, 40014, 6809, 11816, 6479, 12798, 3185, 37400, 6355

Offset

1,2

Comments

Multiplicative because A341529 is.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization.

Index entries for sequences related to sigma(n).

Formula

a(n) = Sum_{d|n} A008683(n/d) * A341529(d).

a(n) = A346239(n) + A347124(n).

Multiplicative with a(p^e) = q^(e-1)*(p^e*(q*p-1)-q+1)/(p-1), where q = A151800(p). - Sebastian Karlsson, Sep 02 2021

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (1/zeta(3)) / Product_{p prime} ((p^2-q)*(p^3-q))/(p^4*(p-1)) = 7.6530842... , and q(p) = A151800(p). - Amiram Eldar, Dec 24 2023

Mathematica

f[p_, e_] := Module[{q = NextPrime[p]}, q^(e-1) * (p^e * (q*p-1)-q+1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2023 *)

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341529(n) = (sigma(n)*A003961(n));
A347125(n) = sumdiv(n, d, moebius(n/d)*A341529(d))

Crossrefs

Cf. A002117, A003961, A003973, A008683, A151800, A341529, A341512, A346239, A347124.

Sequence in context: A359265 A153026 A297302 * A057452 A091560 A061877

Adjacent sequences: A347122 A347123 A347124 * A347126 A347127 A347128

Keyword

nonn,mult

Author

Antti Karttunen, Aug 24 2021

Status

approved