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A323444 Sum of exponents in prime-power factorization of Product_{k=0..n} binomial(n,k) (A001142). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Also sum of exponents in prime-power factorization of hyperfactorial(n) / superfactorial(n).
LINKS
Jeffrey C. Lagarias, Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.
Eric Weisstein's World of Mathematics, Hyperfactorial
Eric Weisstein's World of Mathematics, Superfactorial
FORMULA
a(n) = A303281(n) - A303279(n), for n > 0.
a(n) = A001222(A001142(n)).
EXAMPLE
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.
MATHEMATICA
Array[Sum[PrimeOmega@ Binomial[#, k], {k, 0, #}] &, 57] (* Michael De Vlieger, Jan 19 2019 *)
PROG
(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
CROSSREFS
Sequence in context: A290701 A200579 A200809 * A099027 A306925 A134466
KEYWORD
nonn
AUTHOR
Daniel Suteu, Jan 15 2019
STATUS
approved

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Last modified June 15 03:12 EDT 2024. Contains 373402 sequences. (Running on oeis4.)