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Number of factorizations of n into factors > 1 with odd greatest factor.
10

%I #12 Dec 13 2021 16:14:54

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

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

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

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

%H Antti Karttunen, <a href="/A340831/b340831.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) (4*27) (135) (4*45) (4*63)

%e (5*9) (2*6*9) (3*45) (12*15) (12*21)

%e (3*15) (3*4*9) (5*27) (4*5*9) (4*7*9)

%e (3*3*5) (2*2*27) (9*15) (2*2*45) (6*6*7)

%e (2*2*3*9) (3*5*9) (2*6*15) (2*2*63)

%e (2*2*3*3*3) (3*3*15) (3*4*15) (2*6*21)

%e (3*3*3*5) (2*2*5*9) (3*4*21)

%e (3*3*4*5) (2*2*7*9)

%e (2*2*3*15) (2*3*6*7)

%e (2*2*3*3*5) (3*3*4*7)

%e (2*2*3*21)

%e (2*2*3*3*7)

%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@*Max]],{n,100}]

%o (PARI) A340831(n, m=n, fc=1) = if(1==n, !fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(!fc||(d%2)), s += A340831(n/d, d, 0*fc))); (s)); \\ _Antti Karttunen_, Dec 13 2021

%Y Positions of 0's are A000079.

%Y The version for partitions is A027193.

%Y The version for prime indices is A244991.

%Y The version looking at length instead of greatest factor is A339890.

%Y The version that also has odd length is A340607.

%Y The version looking at least factor is A340832.

%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 A340653 counts balanced factorizations.

%Y - Odd -

%Y A000009 counts partitions into odd parts.

%Y A024429 counts set partitions of odd length.

%Y A026424 lists numbers with odd Omega.

%Y A058695 counts partitions of odd numbers.

%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 A340692 counts partitions of odd rank.

%Y Cf. A026804, A050320, A061395, A339846, A340385, A340596, A340599, A340654, A340655, A340785, A340854, A340855.

%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