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A182860
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Number of distinct prime signatures represented among the unitary divisors of n.
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8
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1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 4, 2, 3, 2, 4, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 4, 2, 4, 3, 4, 2, 4, 2, 3, 4, 4, 3, 4, 2, 4, 2, 3, 2, 6, 3, 3, 3, 4, 2, 6, 3, 4, 3, 3, 3, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4
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OFFSET
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1,2
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COMMENTS
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a(n) = number of members m of A025487 such that d(m^k) divides d(n^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)
a(n) depends only on prime signature of n (cf. A025487).
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LINKS
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FORMULA
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If the canonical factorization of n into prime powers is Product p^e(p), then the formula for d(n^k) is Product_p (ek + 1). (See also A146289, A146290.)
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EXAMPLE
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60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature. Since a total of 6 distinct prime signatures appear among the unitary divisors of 60, a(60) = 6.
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MATHEMATICA
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Table[Length@ Union@ Map[Sort[FactorInteger[#] /. {p_, e_} /; p > 0 :> If[p == 1, 0, e]] &, Select[Divisors@ n, CoprimeQ[#, n/#] &]], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)
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PROG
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(PARI)
A181819(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); }; \\ From A181819
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CROSSREFS
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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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