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A182860 Number of distinct prime signatures represented among the unitary divisors of n. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Unitary Divisor

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(n) = A000005(A181819(n)).

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.)

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(PARI)

A181819(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); } \\ This function from Michel Marcus, Nov 16 2015

A182860(n) = numdiv(A181819(n)); \\ Antti Karttunen, Jul 19 2017

CROSSREFS

Cf. A034444, A085082, A146289, A146290, A182861, A182862, A212180.

Sequence in context: A196941 A062843 A136164 * A328830 A033947 A069719

Adjacent sequences:  A182857 A182858 A182859 * A182861 A182862 A182863

KEYWORD

nonn

AUTHOR

Matthew Vandermast, Jan 14 2011

STATUS

approved

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Last modified December 12 20:12 EST 2019. Contains 329961 sequences. (Running on oeis4.)