A202535 -id:A202535 - cp's OEIS frontend

A064549 a(n) = n * Product_{primes p|n} p.

1, 4, 9, 8, 25, 36, 49, 16, 27, 100, 121, 72, 169, 196, 225, 32, 289, 108, 361, 200, 441, 484, 529, 144, 125, 676, 81, 392, 841, 900, 961, 64, 1089, 1156, 1225, 216, 1369, 1444, 1521, 400, 1681, 1764, 1849, 968, 675, 2116, 2209, 288, 343, 500, 2601, 1352

Offset: 1

Henry Bottomley, Oct 16 2001

Index of first occurrence of n in A003557. - Franklin T. Adams-Watters, Jul 25 2014

			a(12) = 72 since 12 = 2^2*3 and 12*2*3 = 72.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, Vol. 113, No. 8 (2006), pp. 1732-1745; arXiv preprint, arXiv:math/0509316 [math.NT], 2005-2006.

A permutation of the powerful numbers A001694.

Cf. A003557 (a left inverse), A007947, A057521, A078310, A082695, A202535.

a064549 n = a007947 n * n  -- Reinhard Zumkeller, Jul 23 2013

[n^2/( (&+[Floor(k^n/n)-Floor((k^n - 1)/n) : k in [1..n]]) ): n in [1..50]]; // G. C. Greubel, Nov 02 2018

a:= n -> n * convert(numtheory:-factorset(n), `*`):
seq(a(n),n=1..100); # Robert Israel, Jul 25 2014

a[n_] := n * Times @@ FactorInteger[n][[All, 1]]; Array[a, 100] (* Jean-François Alcover, Feb 17 2017 *)
Table[n*Product[If[PrimeQ[d], d, 1], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Jun 15 2019 *)

popf(n)= { local(f,p=1); f=factor(n); for(i=1, matsize(f)[1], p*=f[i, 1]); return(p) } { for (n=1, 1000, write("b064549.txt", n, " ", n*popf(n)) ) } \\ Harry J. Smith, Sep 18 2009

A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); }; \\ Antti Karttunen, Aug 30 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 24 2020

Multiplicative with a(p^k)=p^(k+1) when k>0.
a(n) = n*A007947(n) = n^2/A003557(n).
Dirichlet convolution of A000027 and A202535. - R. J. Mathar, Dec 20 2011
a(n) = A078310(n) - 1. - Reinhard Zumkeller, Jul 23 2013
A003557(a(n)) = n; a(A003557(n)) = A057521(n). - Antti Karttunen, Aug 30 2018
G.f.: Sum_{k>=1} mu(k)^2*phi(k)*k*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
From Vaclav Kotesovec, Jun 24 2020: (Start)
Dirichlet g.f.: zeta(s-2) * zeta(s-1) * Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = A065463/3 = A065464*Pi^2/18 = 0.234814...
(End)
Sum_{k>=1} 1/a(k) = zeta(2)*zeta(3)/zeta(6) = A082695. - Vaclav Kotesovec, Sep 19 2020
Sum_{k>=1} (-1)^(k+1)/a(k) = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 18 2020

A367865 a(n) = Sum_{d|n} d * phi(d) * mu(d)^2.

Original entry on oeis.org

1, 3, 7, 3, 21, 21, 43, 3, 7, 63, 111, 21, 157, 129, 147, 3, 273, 21, 343, 63, 301, 333, 507, 21, 21, 471, 7, 129, 813, 441, 931, 3, 777, 819, 903, 21, 1333, 1029, 1099, 63, 1641, 903, 1807, 333, 147, 1521, 2163, 21, 43, 63, 1911, 471, 2757, 21, 2331, 129, 2401, 2439

Offset: 1

Wesley Ivan Hurt, Dec 03 2023

Inverse Möbius transform of n * phi(n) * mu(n)^2.

N. J. A. Sloane, Transforms.

Cf. A000010 (phi), A007947 (rad), A008966 (mu^2), A202535.

Table[Sum[d*EulerPhi[d]*MoebiusMu[d]^2, {d, Divisors[n]}], {n, 100}]

a(n) = sumdiv(n, d, if (issquarefree(d), d*eulerphi(d))); \\ Michel Marcus, Dec 04 2023

from math import prod
from sympy import primefactors
def A367865(n): return prod(p*(p-1)+1 for p in primefactors(n)) # Chai Wah Wu, Dec 05 2023

Multiplicative with a(p^e) = p^2 - p + 1. - Amiram Eldar, Dec 04 2023

Sum_{k=1..n} a(k) ~ c * n^3/3, where c = Product_{p prime} (1 - 2/(1+p+p^2)) = 0.51478027457383523467921514707014858470711969900467102074735896602342984... - Vaclav Kotesovec, Dec 05 2023

A380322 a(n) is the sum of exponentially odd divisors of n^2.

Original entry on oeis.org

1, 3, 4, 11, 6, 12, 8, 43, 31, 18, 12, 44, 14, 24, 24, 171, 18, 93, 20, 66, 32, 36, 24, 172, 131, 42, 274, 88, 30, 72, 32, 683, 48, 54, 48, 341, 38, 60, 56, 258, 42, 96, 44, 132, 186, 72, 48, 684, 351, 393, 72, 154, 54, 822, 72, 344, 80, 90, 60, 264, 62, 96, 248

Offset: 1

Amiram Eldar, Jan 20 2025

The number of exponentially odd divisors of n^2 is equal to the number of divisors of n, A000005(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A000290, A005117, A033634, A268335.

Cf. A202535.

f[p_, e_] := (p^(2*e+1) - p)/(p^2 - 1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e+1) - p)/(p^2 - 1) + 1);}

a(n) = A033634(A000290(n)) = A033634(n^2).
a(n) >= A000203(n), with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = (p^(2*e+1) - p)/(p^2 - 1) + 1.
Dirichlet g.f.: zeta(s-2) * zeta(s) * Product_{p prime} (1 - 1/p^(s-2) + 1/p^(s-1)).

cp's OEIS Frontend

A064549 a(n) = n * Product_{primes p|n} p.

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A367865 a(n) = Sum_{d|n} d * phi(d) * mu(d)^2.

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A380322 a(n) is the sum of exponentially odd divisors of n^2.

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Mathematica

PARI

Formula