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

1,2

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.

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

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 *)

Prog

(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

Crossrefs

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

Cf. also A353793.

Sequence in context: A332608 A177464 A316985 * A273315 A301100 A008211

Adjacent sequences: A353786 A353787 A353788 * A353790 A353791 A353792

Keyword

nonn,mult

Author

Antti Karttunen, May 10 2022

Status

approved