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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 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 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 *) 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 *) 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, A013661, A078434. Sequence in context: A283983 A016466 A293669 * A082553 A331736 A344300 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 December 4 11:15 EST 2023. Contains 367560 sequences. (Running on oeis4.)