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A336423
Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).
20
1, 1, 1, 2, 1, 0, 1, 4, 2, 0, 1, 5, 1, 0, 0, 8, 1, 5, 1, 5, 0, 0, 1, 14, 2, 0, 4, 5, 1, 0, 1, 16, 0, 0, 0, 0, 1, 0, 0, 14, 1, 0, 1, 5, 5, 0, 1, 36, 2, 5, 0, 5, 1, 14, 0, 14, 0, 0, 1, 0, 1, 0, 5, 32, 0, 0, 1, 5, 0, 0, 1, 35, 1, 0, 5, 5, 0, 0, 1, 36, 8, 0, 1, 0
OFFSET
1,4
COMMENTS
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.
EXAMPLE
The a(n) chains for n = 4, 8, 12, 16, 24, 32:
4/1 8/1 12/1 16/1 24/1 32/1
4/2/1 8/2/1 12/2/1 16/2/1 24/2/1 32/2/1
8/4/1 12/3/1 16/4/1 24/3/1 32/4/1
8/4/2/1 12/4/1 16/8/1 24/4/1 32/8/1
12/4/2/1 16/4/2/1 24/8/1 32/16/1
16/8/2/1 24/12/1 32/4/2/1
16/8/4/1 24/4/2/1 32/8/2/1
16/8/4/2/1 24/8/2/1 32/8/4/1
24/8/4/1 32/16/2/1
24/12/2/1 32/16/4/1
24/12/3/1 32/16/8/1
24/12/4/1 32/8/4/2/1
24/8/4/2/1 32/16/4/2/1
24/12/4/2/1 32/16/8/2/1
32/16/8/4/1
32/16/8/4/2/1
MATHEMATICA
strchns[n_]:=If[n==1, 1, If[!UnsameQ@@Last/@FactorInteger[n], 0, Sum[strchns[d], {d, Select[Most[Divisors[n]], UnsameQ@@Last/@FactorInteger[#]&]}]]];
Table[strchns[n], {n, 100}]
CROSSREFS
A336569 is the maximal case.
A336571 does not require n itself to have distinct prime multiplicities.
A000005 counts divisors.
A007425 counts divisors of divisors.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty strict chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A337256 counts strict chains of divisors.
Sequence in context: A127709 A343730 A343761 * A128307 A349394 A034369
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 27 2020
STATUS
approved