A182861 Number of distinct prime signatures represented among the unitary divisors of A025487(n).

1, 2, 2, 3, 2, 4, 2, 4, 4, 2, 3, 4, 6, 2, 4, 4, 6, 2, 4, 6, 4, 5, 3, 6, 2, 4, 8, 4, 8, 4, 6, 2, 4, 8, 4, 8, 4, 4, 6, 2, 6, 4, 9, 3, 8, 4, 8, 4, 6, 6, 2, 8, 4, 6, 12, 4, 8, 4, 8, 4, 6, 6, 2, 8, 4, 10, 12, 4, 6, 8, 4, 8, 6, 8, 4, 6, 9, 6, 3, 2, 8, 4, 10, 12, 4

Offset

1,2

Comments

a(n) = number of members m of A025487 such that d(m^k) divides d(A025487(n)^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Unitary Divisor

Formula

a(n) = A000005(A181820(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, and since 60 = A025487(13), a(13) = 6.

Crossrefs

Cf. A000005, A025487, A034444, A085082, A146289, A146290, A181820, A182860, A182862.

Sequence in context: A218703 A326641 A144372 * A049238 A001616 A257599

Adjacent sequences: A182858 A182859 A182860 * A182862 A182863 A182864

Keyword

nonn

Author

Matthew Vandermast, Jan 14 2011

Extensions

More terms from Amiram Eldar, Jun 20 2019

Status

approved