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 A057723 Sum of positive divisors of n that are divisible by every prime that divides n. 43
 1, 2, 3, 6, 5, 6, 7, 14, 12, 10, 11, 18, 13, 14, 15, 30, 17, 24, 19, 30, 21, 22, 23, 42, 30, 26, 39, 42, 29, 30, 31, 62, 33, 34, 35, 72, 37, 38, 39, 70, 41, 42, 43, 66, 60, 46, 47, 90, 56, 60, 51, 78, 53, 78, 55, 98, 57, 58, 59, 90, 61, 62, 84, 126, 65, 66, 67, 102, 69, 70 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Ivan Neretin, Table of n, a(n) for n = 1..10000 FORMULA If n = Product p_i^e_i then a(n) = Product (p_i + p_i^2 + ... + p_i^e_i). a(n) = rad(n)*sigma(n/rad(n)) = A007947(n)*A000203(A003557(n)). - Ivan Neretin, May 13 2015 Dirichlet g.f.: zeta(s) * zeta(s-1) * Product(p prime, 1 - p^(-s) + p^(1-2*s)). - Robert Israel, May 13 2015 Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A330596 = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.7485352596823635646442150486379106016416403430053244045... - Vaclav Kotesovec, Dec 18 2019 a(n) = Sum_{d|n, rad(d)=rad(n)} d. - R. J. Mathar, Jun 02 2020 Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Jun 10 2020 a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * sigma(d). - Ilya Gutkovskiy, Apr 15 2021 EXAMPLE The divisors of 12 that are divisible by both 2 and 3 are 6 and 12. So a(12) = 6 + 12 = 18. MAPLE seq(mul(f[1]*(f[1]^f[2]-1)/(f[1]-1), f = ifactors(n)[2]), n = 1 .. 100); # Robert Israel, May 13 2015 MATHEMATICA Table[(b = Times @@ FactorInteger[n][[All, 1]])*DivisorSigma[1, n/b], {n, 70}] (* Ivan Neretin, May 13 2015 *) f[p_, e_] := (p^(e+1)-1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *) PROG (Magma) [&*PrimeDivisors(n)*SumOfDivisors(n div &*PrimeDivisors(n)): n in [1..70]]; // Vincenzo Librandi, May 14 2015 (PARI) a(n) = {my(f = factor(n)); for (i=1, #f~, f[i, 2]=1); my(pp = factorback(f)); sumdiv(n, d, if (! (d % pp), d, 0)); } \\ Michel Marcus, May 14 2015 CROSSREFS Row sums of triangle A284318. Cf. A000203 (sigma), A007947 (rad), A005361 (number of these divisors). Cf. A049060 and A060640 (other sigma-like functions). Cf. A065487, A330596. Sequence in context: A336465 A340774 A345068 * A335835 A361480 A142151 Adjacent sequences: A057720 A057721 A057722 * A057724 A057725 A057726 KEYWORD nonn,easy,mult AUTHOR Leroy Quet, Oct 27 2000 STATUS approved

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Last modified August 2 22:39 EDT 2024. Contains 374875 sequences. (Running on oeis4.)