A204617 Multiplicative with a(p^e) = p^(e-1)*H(p). H(2) = 1, H(p) = p - 1 if p = 1 (mod 4) and H(p) = p + 1 if p = 3 (mod 4).

1, 1, 4, 2, 4, 4, 8, 4, 12, 4, 12, 8, 12, 8, 16, 8, 16, 12, 20, 8, 32, 12, 24, 16, 20, 12, 36, 16, 28, 16, 32, 16, 48, 16, 32, 24, 36, 20, 48, 16, 40, 32, 44, 24, 48, 24, 48, 32, 56, 20, 64, 24, 52, 36, 48, 32, 80, 28, 60, 32, 60, 32, 96, 32, 48, 48, 68, 32

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = phi(n) if n is in A072437.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2) * Product_{primes p == 3 (mod 4)} (1 + 1/p^2) = 3*A243381/(8*A175647) = 0.409404... . - Amiram Eldar, Dec 24 2022

a(n) = n*Product_{primes p, p | n} (1 - A034947(p)/p) = Sum_{d | n} A034947(d)* mobius(d)*n/d. Cf. A000010(n) = Sum_{d | n} mobius(d)*n/d. - Peter Bala, Dec 26 2023

Maple

with(numtheory):
a := n->add(jacobi(-1, d)*mobius(d)*n/d, d in divisors(n)):
seq(a(n), n = 1..60); # Peter Bala, Dec 26 2023

Mathematica

ar[p_, s_] := Which[Mod[p, 4]==1, p^(s-1)*(p-1), Mod[p, 4]==3, p^(s-1)*(p+1), True, p^(s-1)]; arit[1] = 1; arit[n_] := Product[ar[FactorInteger[n][[i, 1]], FactorInteger[n][[i, 2]]], {i, Length[FactorInteger[n]]}]; Array[arit, 100]

Prog

(PARI) A204617(n) = { my(f=factor(n), p); prod(i=1, #f~, p=f[i, 1]; (p^(f[i, 2]-1)) * if(2==p, 1, if(1==(p%4), p-1, p+1))); }; \\ Antti Karttunen, Nov 16 2021

Crossrefs

Cf. A000010, A072437.

Cf. A175647, A243381.

Sequence in context: A057696 A057697 A019921 * A061469 A222641 A368437

Adjacent sequences: A204614 A204615 A204616 * A204618 A204619 A204620

Keyword

nonn,mult

Author

José María Grau Ribas, Jan 17 2012

Status

approved