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A340607 Number of factorizations of n into an odd number of factors > 1, the greatest of which is odd.

%I A340607 #10 Dec 13 2021 16:14:38
%S A340607 0,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,1,1,0,1,0,1,0,2,1,1,1,1,0,1,0,
%T A340607 1,1,1,0,1,1,1,1,1,1,2,0,1,1,1,1,1,1,1,1,1,1,1,0,1,2,1,0,2,0,1,1,1,1,
%U A340607 1,1,1,2,1,0,2,1,1,1,1,2,2,0,1,3,1,0,1,1,1,2,1,1,1,0,1,0,1,1,2,2,1,1,1,1,2,0,1,4
%N A340607 Number of factorizations of n into an odd number of factors > 1, the greatest of which is odd.
%H A340607 Antti Karttunen, <a href="/A340607/b340607.txt">Table of n, a(n) for n = 1..20000</a>
%e A340607 The a(n) factorizations for n = 27, 84, 108, 180, 252, 360, 432:
%e A340607   27     2*6*7   2*6*9      4*5*9      4*7*9      5*8*9       6*8*9
%e A340607   3*3*3  3*4*7   3*4*9      2*2*45     6*6*7      2*4*45      2*8*27
%e A340607          2*2*21  2*2*27     2*6*15     2*2*63     3*8*15      4*4*27
%e A340607                  2*2*3*3*3  3*4*15     2*6*21     4*6*15      2*2*2*6*9
%e A340607                             2*2*3*3*5  3*4*21     2*12*15     2*2*3*4*9
%e A340607                                        2*2*3*3*7  2*2*2*5*9   2*2*2*2*27
%e A340607                                                   2*3*3*4*5   2*2*2*2*3*3*3
%e A340607                                                   2*2*2*3*15
%t A340607 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340607 Table[Length[Select[facs[n],OddQ[Length[#]]&&OddQ[Max@@#]&]],{n,100}]
%o A340607 (PARI) A340607(n, m=n, k=0, grodd=0) = if(1==n, k, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(grodd||(d%2)), s += A340607(n/d, d, 1-k, bitor(1,grodd)))); (s)); \\ _Antti Karttunen_, Dec 13 2021
%Y A340607 Note: Heinz numbers are given in parentheses below.
%Y A340607 The case of odd length only is A339890.
%Y A340607 The case of all odd factors is A340102.
%Y A340607 The version for partitions is A340385.
%Y A340607 The version for prime indices is A340386.
%Y A340607 The case of odd maximum only is A340831.
%Y A340607 A000009 counts partitions into odd parts (A066208).
%Y A340607 A001055 counts factorizations, with strict case A045778.
%Y A340607 A027193 counts partitions of odd length/maximum (A026424/A244991).
%Y A340607 A058695 counts partitions of odd numbers (A300063).
%Y A340607 A078408 counts odd-length partitions into odd numbers (A300272).
%Y A340607 A316439 counts factorizations by sum and length.
%Y A340607 A340101 counts factorizations (into odd factors = of odd numbers).
%Y A340607 A340832 counts factorizations whose least part is odd.
%Y A340607 A340854/A340855 lack/have a factorization with odd minimum.
%Y A340607 Cf. A000700, A024429, A026804, A028260, A061395, A112798, A160786, A236914, A324522, A326845, A340608, A340788.
%K A340607 nonn
%O A340607 1,27
%A A340607 _Gus Wiseman_, Jan 25 2021
%E A340607 Data section extended up to 108 terms by _Antti Karttunen_, Dec 13 2021