A353791 Multiplicative with a(p^e) = ((p-1)*q)^e, where q is the largest prime less than p, and 1 if p = 2.

1, 1, 4, 1, 12, 4, 30, 1, 16, 12, 70, 4, 132, 30, 48, 1, 208, 16, 306, 12, 120, 70, 418, 4, 144, 132, 64, 30, 644, 48, 870, 1, 280, 208, 360, 16, 1116, 306, 528, 12, 1480, 120, 1722, 70, 192, 418, 1978, 4, 900, 144, 832, 132, 2444, 64, 840, 30, 1224, 644, 3074, 48, 3540, 870, 480, 1, 1584, 280, 4026, 208, 1672, 360

Offset: 1

Author

Antti Karttunen, May 11 2022

Keywords

• nonn
• mult

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. A003958, A064989, A151799, A353792 [= a(A000203(n))], A353793 [= a(A003961(n))], A353794 [= a(sigma(A003961(n)))].

Cf. also A353749.

Programs

Mathematica

f[p_, e_] := (If[p == 2, 1, NextPrime[p, -1]]*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, Dec 31 2022 *)

PARI

A353791(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = (f[i,1]-1)*precprime(f[i,1]-1)); factorback(f); };

Formula

a(n) = A003958(n) * A064989(n).

a(n) = a(2*n) = a(A000265(n)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^3-p^2)/(p^3-p*q+q)) = 0.1075035014..., where q(p) = prevprime(p) = A151799(p) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 31 2022