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A333557 a(n) = Sum_{d|n, gcd(d, n/d) = 1} uphi(d) * uphi(n/d), where uphi = unitary totient function (A047994). 0
1, 2, 4, 6, 8, 8, 12, 14, 16, 16, 20, 24, 24, 24, 32, 30, 32, 32, 36, 48, 48, 40, 44, 56, 48, 48, 52, 72, 56, 64, 60, 62, 80, 64, 96, 96, 72, 72, 96, 112, 80, 96, 84, 120, 128, 88, 92, 120, 96, 96, 128, 144, 104, 104, 160, 168, 144, 112, 116, 192, 120, 120, 192, 126, 192, 160, 132, 192, 176, 192 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..70.

FORMULA

If n = Product (p_j^k_j) then a(n) = Product (2 * (p_j^k_j - 1)).

a(n) = 2^omega(n) * uphi(n).

a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-2)^omega(n/d) * 2^omega(d) * d.

a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * A145388(d).

MATHEMATICA

uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[If[GCD[d, n/d] == 1, uphi[d] uphi[n/d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

Table[Sum[If[GCD[d, n/d] == 1, (-2)^PrimeNu[n/d] 2^PrimeNu[d] d, 0], {d, Divisors[n]}], {n, 1, 70}]

PROG

(PARI) a(n) = sumdiv(n, d, if (gcd(d, n/d) == 1, (-2)^omega(n/d)*2^omega(d)*d)); \\ Michel Marcus, Mar 27 2020

CROSSREFS

Cf. A001221, A029935, A034444, A047994, A055653, A062355, A145388.

Sequence in context: A244363 A063199 A219028 * A062355 A087671 A088308

Adjacent sequences:  A333554 A333555 A333556 * A333558 A333559 A333560

KEYWORD

nonn,mult

AUTHOR

Ilya Gutkovskiy, Mar 26 2020

STATUS

approved

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Last modified July 13 08:26 EDT 2020. Contains 335685 sequences. (Running on oeis4.)