A322360 Multiplicative with a(p^e) = p^2 - 1.

1, 3, 8, 3, 24, 24, 48, 3, 8, 72, 120, 24, 168, 144, 192, 3, 288, 24, 360, 72, 384, 360, 528, 24, 24, 504, 8, 144, 840, 576, 960, 3, 960, 864, 1152, 24, 1368, 1080, 1344, 72, 1680, 1152, 1848, 360, 192, 1584, 2208, 24, 48, 72, 2304, 504, 2808, 24, 2880, 144, 2880, 2520, 3480, 576, 3720, 2880, 384, 3, 4032, 2880

Offset: 1

Author

Antti Karttunen, Dec 04 2018

Keywords

• nonn
• mult

Comments

Absolute values of A046970, the Dirichlet inverse of the Jordan function J_2 (A007434).

Absolute values of the Möbius transform of A055491. (See Benoit Cloitre's May 31 2002 comment in A046970).

Links

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

Crossrefs

Absolute values of A046970.

Cf. A048250, A173557, A066086, A322359, A351265.

Programs

Maple

a:= n-> mul(i[1]^2-1, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021

Mathematica

a[n_] := If[n==1, 1, Times @@ ((#^2-1)& @@@ FactorInteger[n])]; Array[a, 50]  (* Amiram Eldar, Dec 05 2018 *)

PARI

A322360(n) = factorback(apply(p -> (p*p)-1, factor(n)[, 1]));

PARI

A322360(n) = abs(sumdiv(n,d,moebius(n/d)*(core(d)^2)));

Formula

Multiplicative with a(p^e) = p^2 - 1.

a(n) = Product_{p prime divides n} (p^2 - 1).

a(n) = abs(A046970(n)).

a(n) = A048250(n) * A173557(n) = A066086(n) * A322359(n).

G.f. for a signed version of the sequence: Sum_{n >= 1} mu(n)*n^2*x^n/(1 - x^n) = Sum_{n >= 1} (-1)^omega(n)*a(n)*x^n = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + 72*x^10 - ..., where mu(n) is the Möbius function A008683(n) and omega(n) = A001221(n) is the number of distinct primes dividing n. - Peter Bala, Mar 05 2022

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (p-1)*(p^2 + 2*p + 2)/(p*(p^2 + p + 1)) = 0.187556464... . - Amiram Eldar, Oct 22 2022

a(n) = A007434(A007947(n)). - Enrique Pérez Herrero, Oct 14 2024