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Number of factorizations of n into factors > 1 with odd least factor.
15

%I #9 Dec 13 2021 16:15:05

%S 0,0,1,0,1,0,1,0,2,0,1,1,1,0,2,0,1,1,1,0,2,0,1,1,2,0,3,0,1,2,1,0,2,0,

%T 2,2,1,0,2,1,1,1,1,0,4,0,1,2,2,1,2,0,1,2,2,1,2,0,1,3,1,0,4,0,2,1,1,0,

%U 2,2,1,3,1,0,4,0,2,1,1,1,5,0,1,3,2,0,2,0,1,5,2,0,2,0,2,2,1,1,4,1,1,1,1,0,5,0,1,6

%N Number of factorizations of n into factors > 1 with odd least factor.

%H Antti Karttunen, <a href="/A340832/b340832.txt">Table of n, a(n) for n = 1..20000</a>

%e The a(n) factorizations for n = 45, 108, 135, 180, 252:

%e (45) (3*36) (135) (3*60) (3*84)

%e (5*9) (9*12) (3*45) (5*36) (7*36)

%e (3*15) (3*4*9) (5*27) (9*20) (9*28)

%e (3*3*5) (3*6*6) (9*15) (5*6*6) (3*3*28)

%e (3*3*12) (3*5*9) (3*3*20) (3*4*21)

%e (3*3*3*4) (3*3*15) (3*4*15) (3*6*14)

%e (3*3*3*5) (3*5*12) (3*7*12)

%e (3*6*10) (3*3*4*7)

%e (3*3*4*5)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],OddQ@*Min]],{n,100}]

%o (PARI) A340832(n, m=n, fc=1) = if(1==n, (m%2)&&!fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A340832(n/d, d, 0*fc))); (s)); \\ _Antti Karttunen_, Dec 13 2021

%Y Positions of 0's are A340854.

%Y Positions of nonzero terms are A340855.

%Y The version for partitions is A026804.

%Y Odd-length factorizations are counted by A339890.

%Y The version looking at greatest factor is A340831.

%Y - Factorizations -

%Y A001055 counts factorizations.

%Y A045778 counts strict factorizations.

%Y A316439 counts factorizations by product and length.

%Y A340101 counts factorizations into odd factors, odd-length case A340102.

%Y A340607 counts factorizations with odd length and greatest factor.

%Y A340653 counts balanced factorizations.

%Y - Odd -

%Y A000009 counts partitions into odd parts.

%Y A026424 lists numbers with odd Omega.

%Y A027193 counts partitions of odd length.

%Y A058695 counts partitions of odd numbers (A300063).

%Y A066208 lists numbers with odd-indexed prime factors.

%Y A067659 counts strict partitions of odd length (A030059).

%Y A174726 counts ordered factorizations of odd length.

%Y A244991 lists numbers whose greatest prime index is odd.

%Y A340692 counts partitions of odd rank.

%Y Cf. A050320, A160786, A340385, A340596, A340599, A340654, A340655, A340931.

%K nonn

%O 1,9

%A _Gus Wiseman_, Feb 04 2021

%E Data section extended up to 108 terms by _Antti Karttunen_, Dec 13 2021