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A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

%I A182938 #35 Mar 28 2025 13:10:09
%S A182938 1,2,3,1,5,6,7,0,3,10,11,3,13,14,15,0,17,6,19,5,21,22,23,0,10,26,1,7,
%T A182938 29,30,31,0,33,34,35,3,37,38,39,0,41,42,43,11,15,46,47,0,21,20,51,13,
%U A182938 53,2,55,0,57,58,59,15,61,62,21,0,65,66
%N A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).
%H A182938 Reinhard Zumkeller, <a href="/A182938/b182938.txt">Table of n, a(n) for n = 1..10000</a>
%H A182938 Vaclav Kotesovec, <a href="/A182938/a182938.jpg">Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000</a>
%F A182938 a(A185359(n)) = 0. - _Reinhard Zumkeller_, Feb 18 2012
%F A182938 Dirichlet g.f.: Product_{p prime} (1 + p^(-s))^p. - _Ilya Gutkovskiy_, Oct 26 2019
%F A182938 Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.33754... - _Vaclav Kotesovec_, Mar 28 2025
%p A182938 A182938 := proc(n) local e,j; e := ifactors(n)[2]:
%p A182938 mul (binomial(e[j][1], e[j][2]), j=1..nops(e)) end:
%p A182938 seq (A182938(n), n=1..100);
%t A182938 a[n_] := Times @@ (Map[Binomial @@ # &, FactorInteger[n], 1]);
%t A182938 Table[a[n], {n, 1, 100}] (* _Kellen Myers_, Jan 16 2011 *)
%o A182938 (PARI) a(n)=prod(i=1,#n=factor(n)~,binomial(n[1,i],n[2,i])) \\ _M. F. Hasler_
%o A182938 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)^p)[n], ", ")) \\ _Vaclav Kotesovec_, Mar 28 2025
%o A182938 (Haskell)
%o A182938 a182938 n = product $ zipWith a007318'
%o A182938    (a027748_row n) (map toInteger $ a124010_row n)
%o A182938 -- _Reinhard Zumkeller_, Feb 18 2012
%Y A182938 Cf. A000026, A001414, A008473, A008474, A008475, A008476, A008477, A028310, A069799.
%Y A182938 Cf. A027748, A124010, A007318, A328745.
%K A182938 nonn,mult
%O A182938 1,2
%A A182938 _Peter Luschny_, Jan 16 2011
%E A182938 Given terms checked with new PARI code by _M. F. Hasler_, Jan 16 2011