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 A157612 Number of factorizations of n! into distinct factors. 9
 1, 1, 1, 2, 5, 16, 57, 253, 1060, 5285, 28762, 191263, 1052276, 8028450, 56576192, 424900240, 2584010916, 24952953943, 178322999025, 1886474434192, 15307571683248, 143131274598786, 1423606577935925, 17668243239613767, 137205093278725072, 1399239022852163764, 15774656316828338767 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The number of factorizations of (n+1)! into k distinct factors can be arranged into the following triangle: 2! 1; 3! 1, 1; 4! 1, 3, 1; 5! 1, 7, 7, 1; ... LINKS Table of n, a(n) for n=0..26. FORMULA a(n) = A045778(A000142(n)). EXAMPLE 3! = 6 = 2*3. a(3) = 2 because there are 2 factorizations of 3!. 4! = 24 = 2*12 = 3*8 = 4*6 = 2*3*4. a(4) = 5 because there are 5 factorizations of 4!. 5! = 120 (1) 5! = 2*60 = 3*40 = 4*30 = 5*24 = 6*20 = 8*15 = 10*12 (7) 5! = 2*3*20 = 2*4*15 = 2*5*12 = 2*6*10 = 3*4*10 = 3*5*8 = 4*5*6 (7) 5! = 2*3*4*5 (1) a(5) = 16 because there are 16 factorizations of 5!. MAPLE with(numtheory): b:= proc(n, k) option remember; `if`(n>k, 0, 1) +`if`(isprime(n), 0, add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n})) end: a:= n-> b(n!\$2): seq(a(n), n=0..12); # Alois P. Heinz, May 26 2013 MATHEMATICA b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d-1]], {d, Divisors[n] ~Complement~ {1, n}}]]; a[n_] := b[n!, n!]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 21 2017, after Alois P. Heinz *) PROG (PARI) \\ See A318286 for count. a(n)={if(n<=1, 1, count(factor(n!)[, 2]))} \\ Andrew Howroyd, Feb 01 2020 CROSSREFS Cf. A076716, A157017, A157229, A318286. See A157836 for continuation of triangle. Sequence in context: A357580 A192635 A009225 * A348103 A184943 A286946 Adjacent sequences: A157609 A157610 A157611 * A157613 A157614 A157615 KEYWORD nonn AUTHOR Jaume Oliver Lafont, Mar 03 2009 EXTENSIONS a(8)-a(12) from Ray Chandler, Mar 07 2009 a(13)-a(17) from Alois P. Heinz, May 26 2013 a(18)-a(19) from Alois P. Heinz, Jan 10 2015 a(20)-a(26) from Andrew Howroyd, Feb 01 2020 STATUS approved

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Last modified September 18 20:35 EDT 2024. Contains 376002 sequences. (Running on oeis4.)