A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.

1, 4, 12, 24, 30, 48, 70, 144, 180, 120, 132, 288, 208, 280, 360, 864, 306, 720, 418, 720, 840, 528, 644, 1728, 1050, 832, 2700, 1680, 870, 1440, 1116, 5184, 1584, 1224, 2100, 4320, 1480, 1672, 2496, 4320, 1722, 3360, 1978, 3168, 5400, 2576, 2444, 10368, 5390, 4200, 3672, 4992, 3074, 10800, 3960, 10080, 5016, 3480

Offset: 1

Author

Antti Karttunen, May 10 2022

Keywords

• nonn
• mult

Comments

Question: Does a(n) divide A353790(n) only when n=1? Compare to A353764.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

Cf. A003961, A003972, A151800, A353749, A353764, A353790.

Cf. also A353793.

Programs

Mathematica

f[p_, e_] := ((q = NextPrime[p]) - 1) * q^(e - 1) * p^e; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 10 2022 *)

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A353789(n) = (n * eulerphi(A003961(n)));

Python

from math import prod
from sympy import nextprime, factorint
def A353789(n): return prod((q:= nextprime(p))**(e-1)*p**e*(q-1) for p, e in factorint(n).items()) # Chai Wah Wu, May 10 2022

Formula

Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.

a(n) = A353749(A003961(n)) = n * A003972(n).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^3-p^2-p+1)/(p^3 - p*q)) = 0.836506229..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022