OFFSET
0,1
COMMENTS
Number of ways to factor p^n*q^2 where p and q are distinct primes.
a(n) is the number of multiset partitions of the multiset {r^n, s^2}. - Joerg Arndt, Jan 01 2024
REFERENCES
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
Amarnath Murthy, "Generalization of Smarandache Factor Partition introducing Smarandache Factor Partition". Smarandache Notions Journal, 1-2-3, vol. 11, 2000.
Amarnath Murthy, Program for finding out the number of Smarandache Factor Partitions. Smarandache Notions Journal, Vol. 13, 2002.
Amarnath Murthy, e-book, MS LIT format, "Ideas on Smarandache Notions".
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.9, 1.14.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers, National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956. (Annotated scanned pages from, plus a review)
FORMULA
From Vaclav Kotesovec, Feb 01 2016, corrected Nov 05 2016: (Start)
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (4*Pi^2) * (1 + 83*Pi/(24*sqrt(6*n))).
(End)
EXAMPLE
a(2) = 9: let p = 2 and q = 3, p^2*q^2 = 36; there are 9 factorizations: (36), (18*2), (12*3), (9*4), (9*2^2), (6*6), (6*3*2), (4*3^2), (3^2*2^2).
MATHEMATICA
max = 40; col = 2; s1 = Series[Product[1/(1-x^(n-k)*y^k), {n, 1, max+2}, {k, 0, n}], {y, 0, col}] // Normal; s2 = Series[s1, {x, 0, max+1}]; a[n_] := SeriesCoefficient[s2, {x, 0, n}, {y, 0, col}]; Table[ a[n] , {n, 0, max}] (* Jean-François Alcover, Mar 13 2014 *)
nmax = 50; CoefficientList[Series[1/(1-x)*(1 + 1/(1-x^2))*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Christian G. Bower, Jan 08 2004
STATUS
approved