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A340607
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Number of factorizations of n into an odd number of factors > 1, the greatest of which is odd.
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20
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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
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OFFSET
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1,27
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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Note: Heinz numbers are given in parentheses below.
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.
A316439 counts factorizations by sum and length.
A340101 counts factorizations (into odd factors = of odd numbers).
A340832 counts factorizations whose least part is odd.
Cf. A000700, A024429, A026804, A028260, A061395, A112798, A160786, A236914, A324522, A326845, A340608, A340788.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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