A257464 Number of factorizations of m^n into 3 factors, where m is a product of exactly 3 distinct primes and each factor is a product of n primes (counted with multiplicity).

1, 1, 5, 10, 23, 40, 73, 114, 180, 262, 379, 521, 712, 938, 1228, 1567, 1986, 2469, 3052, 3715, 4499, 5383, 6410, 7558, 8875, 10335, 11991, 13816, 15865, 18110, 20611, 23336, 26350, 29620, 33213, 37095, 41338, 45904, 50870, 56197, 61964, 68131, 74782, 81873

Offset: 0

Author

Alois P. Heinz, Apr 24 2015

Keywords

• nonn
• easy

Examples

			a(2) = 5: (2*3*5)^2 = 900 = 10*10*9 = 15*10*6 = 15*15*4 = 25*6*6 = 25*9*4.
a(4) = 23: (2*3*5)^4 = 810000 = 100*90*90 = 100*100*81 = 135*100*60 = 150*90*60 = 150*100*54 = 150*135*40 = 150*150*36 = 225*60*60 = 225*90*40 = 225*100*36 = 225*150*24 = 225*225*16 = 250*60*54 = 250*81*40 = 250*90*36 = 250*135*24 = 375*54*40 = 375*60*36 = 375*90*24 = 375*135*16 = 625*36*36 = 625*54*24 = 625*81*16.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Alexander V. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.
• P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. This sequence is column 3 of table f(m,n) on page 40.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Crossrefs

Row n=3 of A257463.

Programs

Maple

a:= n-> coeff(series(-(x^6-x^5+2*x^4+2*x^3+2*x^2-x+1)/
        ((x^2+x+1)*(x+1)^2*(x-1)^5), x, n+1), x, n):
seq(a(n), n=0..60);

Mathematica

CoefficientList[Series[-(x^6 - x^5 + 2 x^4 + 2 x^3 + 2 x^2 - x + 1)/((x^2 + x + 1) (x + 1)^2*(x - 1)^5), {x, 0, 43}], x] (* Michael De Vlieger, Jul 02 2018 *)
LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{1,1,5,10,23,40,73,114,180},50] (* Harvey P. Dale, Jan 08 2022 *)

Formula

G.f.: -(x^6-x^5+2*x^4+2*x^3+2*x^2-x+1)/((x^2+x+1)*(x+1)^2*(x-1)^5).