A055210 Sum of totients of square divisors of n.
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
Offset: 1
Examples
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.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
Programs
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Magma
[&+[EulerPhi(d):d in Divisors(n)| IsSquare(d)]: n in [1..100]]; // Marius A. Burtea, Oct 14 2019
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Mathematica
Array[DivisorSum[#, EulerPhi, IntegerQ@ Sqrt@ # &] &, 97] (* Michael De Vlieger, Nov 18 2017 *) f[p_, e_] := If[EvenQ[e], (p^(e + 1) + 1)/(p + 1), (p^e + 1)/(p + 1)]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 09 2020 *)
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PARI
a(n) = sumdiv(n, d, eulerphi(d)*issquare(d)); \\ Michel Marcus, Dec 31 2013
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
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
G.f.: Sum_{k>=1} k * phi(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 20 2021
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(2)) = 0.529377... . - Amiram Eldar, Nov 13 2022