A353749 a(n) = phi(n) * A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

1, 1, 4, 2, 12, 4, 30, 4, 24, 12, 70, 8, 132, 30, 48, 8, 208, 24, 306, 24, 120, 70, 418, 16, 180, 132, 144, 60, 644, 48, 870, 16, 280, 208, 360, 48, 1116, 306, 528, 48, 1480, 120, 1722, 140, 288, 418, 1978, 32, 1050, 180, 832, 264, 2444, 144, 840, 120, 1224, 644, 3074, 96, 3540, 870, 720, 32, 1584, 280, 4026, 416

Offset: 1

Author

Antti Karttunen, May 07 2022

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

Cf. A000010, A064989, A151800, A353747, A353748, A353750.

Programs

Mathematica

f[p_, e_] := (p - 1)*p^(e - 1)*If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, May 07 2022 *)

PARI

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));

Formula

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

a(n) = A000010(n) * A064989(n).

For n >= 0, a(4n+2) = a(2n+1).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) * Product_{p prime} ((p^3-q)/((p+1)*(p^2-q))) = 0.1118576617..., where q(p) = nextprime(p) = A151800(p). - Amiram Eldar, Dec 31 2022