This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347466 #25 Jul 28 2024 16:39:09
%S A347466 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,
%T A347466 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,
%U A347466 2,269,2,9,29,77,9,66,2,29,9,66,2,323,2,9,29,29
%N A347466 Number of factorizations of n^2.
%C A347466 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%H A347466 Antti Karttunen, <a href="/A347466/b347466.txt">Table of n, a(n) for n = 1..16384</a>
%H A347466 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A347466 a(n) = A001055(A000290(n)).
%e A347466 The a(1) = 1 through a(8) = 11 factorizations:
%e A347466 () (4) (9) (16) (25) (36) (49) (64)
%e A347466 (2*2) (3*3) (2*8) (5*5) (4*9) (7*7) (8*8)
%e A347466 (4*4) (6*6) (2*32)
%e A347466 (2*2*4) (2*18) (4*16)
%e A347466 (2*2*2*2) (3*12) (2*4*8)
%e A347466 (2*2*9) (4*4*4)
%e A347466 (2*3*6) (2*2*16)
%e A347466 (3*3*4) (2*2*2*8)
%e A347466 (2*2*3*3) (2*2*4*4)
%e A347466 (2*2*2*2*4)
%e A347466 (2*2*2*2*2*2)
%p A347466 b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
%p A347466 add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
%p A347466 end:
%p A347466 a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
%p A347466 sort(map(i-> i[2], ifactors(n^2)[2]), `>`))$2)
%p A347466 end:
%p A347466 seq(a(n), n=1..76); # _Alois P. Heinz_, Oct 14 2021
%t A347466 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A347466 Table[Length[facs[n^2]],{n,25}]
%o A347466 (PARI)
%o A347466 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));
%o A347466 A347466(n) = A001055(n^2); \\ _Antti Karttunen_, Oct 13 2021
%Y A347466 Positions of 2's are the primes (A000040), which have squares A001248.
%Y A347466 The restriction to powers of 2 is A058696.
%Y A347466 The additive version (partitions) is A072213.
%Y A347466 The case of integer alternating product is A347459, nonsquared A347439.
%Y A347466 A000290 lists squares, complement A000037.
%Y A347466 A001055 counts factorizations.
%Y A347466 A339846 counts even-length factorizations.
%Y A347466 A339890 counts odd-length factorizations.
%Y A347466 A347050 = factorizations with alternating permutation, complement A347706.
%Y A347466 Cf. A000041, A062312, A120452, A144338, A273013, A330972, A345957, A346635, A347437, A347438, A347457, A347460, A347464.
%K A347466 nonn
%O A347466 1,2
%A A347466 _Gus Wiseman_, Sep 23 2021