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A336571
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Number of sets of divisors d|n, 1 < d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.
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22
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1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 5, 1, 5, 3, 3, 1, 14, 2, 3, 4, 5, 1, 4, 1, 16, 3, 3, 3, 17, 1, 3, 3, 14, 1, 4, 1, 5, 5, 3, 1, 36, 2, 5, 3, 5, 1, 14, 3, 14, 3, 3, 1, 16, 1, 3, 5, 32, 3, 4, 1, 5, 3, 4, 1, 35, 1, 3, 5, 5, 3, 4, 1, 36, 8, 3, 1
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OFFSET
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1,4
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COMMENTS
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A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
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LINKS
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EXAMPLE
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The a(n) sets for n = 4, 6, 12, 16, 24, 84, 36:
{} {} {} {} {} {} {}
{2} {2} {2} {2} {2} {2} {2}
{3} {3} {4} {3} {3} {3}
{4} {8} {4} {4} {4}
{2,4} {2,4} {8} {7} {9}
{2,8} {12} {12} {12}
{4,8} {2,4} {28} {18}
{2,4,8} {2,8} {2,4} {2,4}
{4,8} {2,12} {3,9}
{2,12} {2,28} {2,12}
{3,12} {3,12} {2,18}
{4,12} {4,12} {3,12}
{2,4,8} {4,28} {3,18}
{2,4,12} {7,28} {4,12}
{2,4,12} {9,18}
{2,4,28} {2,4,12}
{3,9,18}
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MATHEMATICA
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strchns[n_]:=If[n==1, 1, Sum[strchns[d], {d, Select[Most[Divisors[n]], UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[strchns[n], {n, 100}]
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CROSSREFS
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A336423 is the version for chains containing n.
A007425 counts divisors of divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A001222, A002033, A005117, A098859, A118914, A124010, A294068, A305149, A327498, A327523, A336568, A336569.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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