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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).
3
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
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
a(n) = A079458(n)/A062570(n). - Ridouane Oudra, Jun 04 2024
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
KEYWORD
nonn,mult
AUTHOR
STATUS
approved