A323444 Sum of exponents in prime-power factorization of Product_{k=0..n} binomial(n,k) (A001142).

0, 0, 1, 2, 6, 6, 11, 10, 23, 28, 33, 28, 45, 38, 44, 50, 86, 74, 96, 82, 106, 110, 114, 96, 147, 150, 153, 182, 211, 184, 215, 186, 281, 280, 279, 278, 347, 308, 306, 304, 380, 336, 374, 328, 368, 408, 403, 352, 489, 482, 524, 516, 559, 498, 596, 586, 686, 674

Offset: 0

Author

Daniel Suteu, Jan 15 2019

Keywords

• nonn

Comments

Also sum of exponents in prime-power factorization of hyperfactorial(n) / superfactorial(n).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
• Eric Weisstein's World of Mathematics, Hyperfactorial.
• Eric Weisstein's World of Mathematics, Superfactorial.
• Index entries for sequences computed from exponents in factorization of n.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A000178, A001142, A001222, A002109, A004788, A007318, A022559, A303279, A303281.

Programs

Mathematica

Array[Sum[PrimeOmega@ Binomial[#, k], {k, 0, #}] &, 57] (* Michael De Vlieger, Jan 19 2019 *)

PARI

a(n) = sum(k=0, n, bigomega(binomial(n, k)));

PARI

a(n) = my(t=0); sum(k=1, n, my(b=bigomega(k)); t+=b; k*b-t);

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

Formula

a(n) = A303281(n) - A303279(n), for n > 0.

a(n) = A001222(A001142(n)).