A111306 - OEIS

A111306

d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).

1, 12, 12, 78, 12, 144, 12, 364, 78, 144, 12, 936, 12, 144, 144, 1365, 12, 936, 12, 936, 144, 144, 12, 4368, 78, 144, 364, 936, 12, 1728, 12, 4368, 144, 144, 144, 6084, 12, 144, 144, 4368, 12, 1728, 12, 936, 936, 144, 12, 16380, 78, 936, 144, 936, 12, 4368, 144 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	LINKS	`Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)` `Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.`
	FORMULA	`G.f.: Sum_{k>=1} tau_11(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018` `Multiplicative with a(p^e) = binomial(e+11,11). - Amiram Eldar, Sep 13 2020`
	MAPLE	b:= proc(n, k) option remember; `if`(k=1, 1, `add(b(d, k-1), d=numtheory[divisors](n)))` `end:` `a:= n-> b(n, 12):` `seq(a(n), n=1..55); # Alois P. Heinz, Jun 12 2024`
	MATHEMATICA	`tau[k_, 1]:=1; tau[k_, n_]:=Times@@(Binomial[#+k-1, k-1]&/@FactorInteger[n][[All, 2]]); Table[tau[12, n], {n, 1000}] (* Enrique Pérez Herrero, Jan 17 2013 *)`
	PROG	`(PARI) for(n=1, 100, print1(sumdiv(n, i, sumdiv(i, j, sumdiv(j, k, sumdiv(k, l, sumdiv(l, m, sumdiv(m, o, sumdiv(o, p, sumdiv(p, q, sumdiv(q, r, sumdiv(r, x, numdiv(x))))))))))), ", "))` `(PARI) a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+11, 11)) \\ Charles R Greathouse IV, Oct 28 2017`
	CROSSREFS	`Cf. tau_2(n)...tau_11(n): A000005, A007425, A007426, A061200, A034695, A111217, A111218, A111219, A111220, A111221.` `Column k=12 of A077592.` `Sequence in context: A003877 A161196 A328531 * A151777 A143478 A219400` `Adjacent sequences: A111303 A111304 A111305 * A111307 A111308 A111309`
	KEYWORD	`mult,nonn`
	AUTHOR	`Gerald McGarvey, Nov 02 2005`
	STATUS	`approved`