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A347466
Number of factorizations of n^2.
8
1, 2, 2, 5, 2, 9, 2, 11, 5, 9, 2, 29, 2, 9, 9, 22, 2, 29, 2, 29, 9, 9, 2, 77, 5, 9, 11, 29, 2, 66, 2, 42, 9, 9, 9, 109, 2, 9, 9, 77, 2, 66, 2, 29, 29, 9, 2, 181, 5, 29, 9, 29, 2, 77, 9, 77, 9, 9, 2, 269, 2, 9, 29, 77, 9, 66, 2, 29, 9, 66, 2, 323, 2, 9, 29, 29
OFFSET
1,2
COMMENTS
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
FORMULA
a(n) = A001055(A000290(n)).
EXAMPLE
The a(1) = 1 through a(8) = 11 factorizations:
() (4) (9) (16) (25) (36) (49) (64)
(2*2) (3*3) (2*8) (5*5) (4*9) (7*7) (8*8)
(4*4) (6*6) (2*32)
(2*2*4) (2*18) (4*16)
(2*2*2*2) (3*12) (2*4*8)
(2*2*9) (4*4*4)
(2*3*6) (2*2*16)
(3*3*4) (2*2*2*8)
(2*2*3*3) (2*2*4*4)
(2*2*2*2*4)
(2*2*2*2*2*2)
MAPLE
b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
end:
a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
sort(map(i-> i[2], ifactors(n^2)[2]), `>`))$2)
end:
seq(a(n), n=1..76); # Alois P. Heinz, Oct 14 2021
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[facs[n^2]], {n, 25}]
PROG
(PARI)
A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A347466(n) = A001055(n^2); \\ Antti Karttunen, Oct 13 2021
CROSSREFS
Positions of 2's are the primes (A000040), which have squares A001248.
The restriction to powers of 2 is A058696.
The additive version (partitions) is A072213.
The case of integer alternating product is A347459, nonsquared A347439.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347050 = factorizations with alternating permutation, complement A347706.
Sequence in context: A305798 A305808 A353858 * A266662 A289078 A051612
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 23 2021
STATUS
approved