A055631 Sum of Euler's totient function phi of distinct primes dividing n.

0, 1, 2, 1, 4, 3, 6, 1, 2, 5, 10, 3, 12, 7, 6, 1, 16, 3, 18, 5, 8, 11, 22, 3, 4, 13, 2, 7, 28, 7, 30, 1, 12, 17, 10, 3, 36, 19, 14, 5, 40, 9, 42, 11, 6, 23, 46, 3, 6, 5, 18, 13, 52, 3, 14, 7, 20, 29, 58, 7, 60, 31, 8, 1, 16, 13, 66, 17, 24, 11, 70, 3, 72, 37, 6, 19, 16, 15, 78, 5, 2

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

a(n) = Sum_{p divides n} p-1 = A008472(n) - A001221(n).

If n = p^w, a power of prime, then a(n) = p-1; if n = 2p, then a(n) = p = n/2.

Additive with a(p^e) = p-1: a(10) = a(2*5) = a(2)+a(5) = (2-1)+(5-1) = 5; a(28) = a(2^2*7) = a(2^2)+a(7) = 1+6 = 7. - Vladeta Jovovic, Oct 23 2001

G.f.: Sum_{k>=1} (prime(k) - 1) * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Aug 18 2021

Maple

with(numtheory); a := n -> add(f, f = map(phi, factorset(n)));
seq(a(n), n = 1..81);  # Peter Luschny, Mar 30 2014

Mathematica

Join[{0}, Table[Total[EulerPhi[Transpose[FactorInteger[n]][[1]]]], {n, 2, 90}]] (* Harvey P. Dale, Oct 29 2012 *)

Prog

(PARI) A055631(n)=sum(i=1, #n=factor(n)~, n[1, i]-1) \\ M. F. Hasler, Nov 10 2016

Crossrefs

Cf. A000010, A001221, A006093, A008472, A053570, A053571, A055632.

Sequence in context: A262663 A075348 A326062 * A234586 A338657 A347082

Adjacent sequences: A055628 A055629 A055630 * A055632 A055633 A055634

Keyword

nonn

Author

Labos Elemer, Jun 06 2000

Extensions

Edited by M. F. Hasler, Nov 10 2016

Status

approved