

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



