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%I #60 Sep 08 2022 08:45:04
%S 1,4,9,8,25,36,49,16,27,100,121,72,169,196,225,32,289,108,361,200,441,
%T 484,529,144,125,676,81,392,841,900,961,64,1089,1156,1225,216,1369,
%U 1444,1521,400,1681,1764,1849,968,675,2116,2209,288,343,500,2601,1352
%N a(n) = n * Product_{primes p|n} p.
%C Index of first occurrence of n in A003557. - _Franklin T. Adams-Watters_, Jul 25 2014
%H Seiichi Manyama, <a href="/A064549/b064549.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%H Nadia Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://dx.doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, Vol. 113, No. 8 (2006), pp. 1732-1745; <a href="https://arxiv.org/abs/math/0509316">arXiv preprint</a>, arXiv:math/0509316 [math.NT], 2005-2006.
%F Multiplicative with a(p^k)=p^(k+1) when k>0.
%F a(n) = n*A007947(n) = n^2/A003557(n).
%F Dirichlet convolution of A000027 and A202535. - _R. J. Mathar_, Dec 20 2011
%F a(n) = A078310(n) - 1. - _Reinhard Zumkeller_, Jul 23 2013
%F A003557(a(n)) = n; a(A003557(n)) = A057521(n). - _Antti Karttunen_, Aug 30 2018
%F G.f.: Sum_{k>=1} mu(k)^2*phi(k)*k*x^k/(1 - x^k)^2. - _Ilya Gutkovskiy_, Nov 02 2018
%F From _Vaclav Kotesovec_, Jun 24 2020: (Start)
%F Dirichlet g.f.: zeta(s-2) * zeta(s-1) * Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = A065463/3 = A065464*Pi^2/18 = 0.234814...
%F (End)
%F Sum_{k>=1} 1/a(k) = zeta(2)*zeta(3)/zeta(6) = A082695. - _Vaclav Kotesovec_, Sep 19 2020
%F Sum_{k>=1} (-1)^(k+1)/a(k) = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - _Amiram Eldar_, Nov 18 2020
%e a(12) = 72 since 12 = 2^2*3 and 12*2*3 = 72.
%p a:= n -> n * convert(numtheory:-factorset(n), `*`):
%p seq(a(n),n=1..100); # _Robert Israel_, Jul 25 2014
%t a[n_] := n * Times @@ FactorInteger[n][[All, 1]]; Array[a, 100] (* _Jean-François Alcover_, Feb 17 2017 *)
%t Table[n*Product[If[PrimeQ[d], d, 1], {d, Divisors[n]}], {n, 1, 100}] (* _Vaclav Kotesovec_, Jun 15 2019 *)
%o (PARI) 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
%o (PARI) A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); }; \\ _Antti Karttunen_, Aug 30 2018
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X)/(1 - p*X))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 24 2020
%o (Haskell)
%o a064549 n = a007947 n * n -- _Reinhard Zumkeller_, Jul 23 2013
%o (Magma) [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
%Y A permutation of the powerful numbers A001694.
%Y Cf. A003557 (a left inverse), A007947, A057521, A078310, A082695, A202535.
%K mult,nonn
%O 1,2
%A _Henry Bottomley_, Oct 16 2001
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