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A055210 Sum of totients of square divisors of n. 2
1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 3, 1, 1, 1, 11, 1, 7, 1, 3, 1, 1, 1, 3, 21, 1, 7, 3, 1, 1, 1, 11, 1, 1, 1, 21, 1, 1, 1, 3, 1, 1, 1, 3, 7, 1, 1, 11, 43, 21, 1, 3, 1, 7, 1, 3, 1, 1, 1, 3, 1, 1, 7, 43, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 21, 3, 1, 1, 1, 11, 61, 1, 1, 3, 1, 1, 1, 3, 1, 7, 1, 3, 1, 1, 1, 11, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

FORMULA

a(n) = Sum_{d is square and divides n} phi(d).

Multiplicative with a(p^e) = (p^(e+1)+1)/(p+1) for even e and a(p^e) = (p^e+1)/(p+1) for odd e. - Vladeta Jovovic, Dec 01 2001

a(n) = Sum_{d|n} A010052(d)*A000010(d). - Antti Karttunen, Nov 18 2017

Conjecture: a(n) = sigma_2(n/core(n))/sigma_1(n/core(n)) = A001157(A008833(n))/A000203(A008833(n)) for all n > 0. - Velin Yanev, Oct 13 2019

EXAMPLE

n = 400: its square divisors are {1, 4, 16, 25, 100, 400}, their totients are {1, 2, 8, 20, 40, 160} and the totient-sum over these divisors is, so a(400) = 231. This value arises at special squarefree multiples of 400 (400 times 2, 3, 5, 6, 7, 10, 11, 13, 15, 17, 19, 21, 22, 23 etc).

a(400) = a(2^4*5^2) = (2^5 + 1)/3*(5^3 + 1)/6 = 231.

MATHEMATICA

Array[DivisorSum[#, EulerPhi, IntegerQ@ Sqrt@ # &] &, 97] (* Michael De Vlieger, Nov 18 2017 *)

PROG

(PARI) a(n) = sumdiv(n, d, eulerphi(d)*issquare(d)); \\ Michel Marcus, Dec 31 2013

(MAGMA) [&+[EulerPhi(d):d in Divisors(n)| IsSquare(d)]: n in [1..100]]; // Marius A. Burtea, Oct 14 2019

CROSSREFS

Cf. A000010, A010052.

Sequence in context: A283983 A016466 A293669 * A082553 A331736 A323840

Adjacent sequences:  A055207 A055208 A055209 * A055211 A055212 A055213

KEYWORD

nonn,mult

AUTHOR

Labos Elemer, Jun 19 2000

STATUS

approved

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Last modified September 24 23:03 EDT 2020. Contains 337325 sequences. (Running on oeis4.)