A347124 Möbius transform of A341528, n * sigma(A003961(n)).

1, 7, 17, 44, 39, 119, 83, 268, 261, 273, 153, 748, 233, 581, 663, 1616, 339, 1827, 455, 1716, 1411, 1071, 689, 4556, 1385, 1631, 3933, 3652, 927, 4641, 1177, 9712, 2601, 2373, 3237, 11484, 1553, 3185, 3961, 10452, 1803, 9877, 2063, 6732, 10179, 4823, 2537, 27472, 6433, 9695, 5763, 10252, 3179, 27531, 5967, 22244

Offset: 1

Author

Antti Karttunen, Aug 24 2021

Keywords

• nonn
• mult

Comments

Multiplicative because A341528 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).

Crossrefs

Cf. A002117, A003961, A003973, A008683, A151800, A341528, A341512, A346239, A347125.

Programs

Mathematica

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

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A003973(n) = sigma(A003961(n));
A341528(n) = (n*A003973(n));
A347124(n) = sumdiv(n,d,moebius(n/d)*A341528(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A341528(d).

a(n) = A347125(n) - A346239(n).

Multiplicative with a(p^e) = p^(e-1)*(q^e*(p*q-1)-p+1)/(q-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} (1 - (q(p)-p)/p^2 - q(p)/p^3) = 5.6488805... , and q(p) = A151800(p). - Amiram Eldar, Dec 24 2023