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Antti Karttunen, <a href="/A340607/b340607.txt">Table of n, a(n) for n = 1..20000</a>

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, 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, 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

(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

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

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Positions of zeros include A340854.

Positions of nonzero terms are included in A340855.

A024429 A027193 counts set partitions of odd length/maximum (A026424/A244991).

A027193 counts partitions of odd length (A026424).

A027193 counts partitions with odd maximum (A244991).

A061395 gives maximum prime indexA078408 counts odd-length partitions into odd numbers (A300272).

A067659 counts strict partitions of odd length.

A160786 counts odd-length partitions of odd numbers (A300272).

A340854 /A340855 lack /have a factorization with odd minimum.

A340855 have a factorization with odd minimum.

Cf. A000700, A024429, A026804, A028260, A078408, A061395, A112798, A174725, A160786, A236914, A316413, A324522, A326845, A340608, A340788.

Note: Heinz numbers are given in parentheses below.

allocated for Gus WisemanNumber of factorizations of n into an odd number of factors > 1, the greatest of which is odd.

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, 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, 1, 1, 1, 2, 1, 0, 2, 1, 1, 1, 1, 2, 2, 0, 1, 3, 1, 0, 1

1,27

The a(n) factorizations for n = 27, 84, 108, 180, 252, 360, 432:

27 2*6*7 2*6*9 4*5*9 4*7*9 5*8*9 6*8*9

3*3*3 3*4*7 3*4*9 2*2*45 6*6*7 2*4*45 2*8*27

2*2*21 2*2*27 2*6*15 2*2*63 3*8*15 4*4*27

2*2*3*3*3 3*4*15 2*6*21 4*6*15 2*2*2*6*9

2*2*3*3*5 3*4*21 2*12*15 2*2*3*4*9

2*2*3*3*7 2*2*2*5*9 2*2*2*2*27

2*3*3*4*5 2*2*2*2*3*3*3

2*2*2*3*15

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

Table[Length[Select[facs[n], OddQ[Length[#]]&&OddQ[Max@@#]&]], {n, 100}]

The case of odd length only is A339890.

The case of all odd factors is A340102.

The version for partitions is A340385.

The version for prime indices is A340386.

The case of odd maximum only is A340831.

Positions of zeros include A340854.

Positions of nonzero terms are included in A340855.

A000009 counts partitions into odd parts (A066208).

A001055 counts factorizations, with strict case A045778.

A024429 counts set partitions of odd length.

A027193 counts partitions of odd length (A026424).

A027193 counts partitions with odd maximum (A244991).

A058695 counts partitions of odd numbers (A300063).

A061395 gives maximum prime index.

A067659 counts strict partitions of odd length.

A160786 counts odd-length partitions of odd numbers (A300272).

A316439 counts factorizations by sum and length.

A340101 counts factorizations (into odd factors = of odd numbers).

A340832 counts factorizations whose least part is odd.

A340854 lack a factorization with odd minimum.

A340855 have a factorization with odd minimum.

Cf. A000700, A026804, A028260, A078408, A112798, A174725, A236914, A316413, A324522, A326845, A340608, A340788.

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Gus Wiseman, Jan 25 2021

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allocated for Gus Wiseman

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