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%I A323444 #28 Feb 01 2019 11:56:52
%S A323444 0,0,1,2,6,6,11,10,23,28,33,28,45,38,44,50,86,74,96,82,106,110,114,96,
%T A323444 147,150,153,182,211,184,215,186,281,280,279,278,347,308,306,304,380,
%U A323444 336,374,328,368,408,403,352,489,482,524,516,559,498,596,586,686,674
%N A323444 Sum of exponents in prime-power factorization of Product_{k=0..n} binomial(n,k) (A001142).
%C A323444 Also sum of exponents in prime-power factorization of hyperfactorial(n) / superfactorial(n).
%H A323444 Jeffrey C. Lagarias, Harsh Mehta, <a href="http://arxiv.org/abs/1409.4145">Products of binomial coefficients and unreduced Farey fractions</a>, arXiv:1409.4145 [math.NT], 2014.
%H A323444 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>
%H A323444 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Superfactorial.html">Superfactorial</a>
%H A323444 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%H A323444 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A323444 a(n) = A303281(n) - A303279(n), for n > 0.
%F A323444 a(n) = A001222(A001142(n)).
%e A323444 a(4) = 6 because C(4,0)*C(4,1)*C(4,2)*C(4,3)*C(4,4) = 2^5 * 3^1 and 5 + 1 = 6, where C(n,k) is the binomial coefficient.
%t A323444 Array[Sum[PrimeOmega@ Binomial[#, k], {k, 0, #}] &, 57] (* _Michael De Vlieger_, Jan 19 2019 *)
%o A323444 (PARI) a(n) = sum(k=0, n, bigomega(binomial(n, k)));
%o A323444 (PARI) a(n) = my(t=0); sum(k=1, n, my(b=bigomega(k)); t+=b; k*b-t);
%o A323444 (PARI) first(n) = my(res = List([0]), r=0, t=0, b=0); for(k=1, n, b=bigomega(k); t += b; r += k*b-t; listput(res, r)); res \\ adapted from _Daniel Suteu_ \\ _David A. Corneth_, Jan 16 2019
%Y A323444 Cf. A000178, A001142, A001222, A002109, A004788, A007318, A022559, A303279, A303281.
%K A323444 nonn
%O A323444 0,4
%A A323444 _Daniel Suteu_, Jan 15 2019

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