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A182861 Number of distinct prime signatures represented among the unitary divisors of A025487(n). 3

%I

%S 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,

%T 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,

%U 4,6,8,4,8,6,8,4,6,9,6,3,2,8,4,10,12,4

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

%C 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).)

%H Amiram Eldar, <a href="/A182861/b182861.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>

%F a(n) = A000005(A181820(n)).

%F 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.)

%e 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.

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

%K nonn

%O 1,2

%A _Matthew Vandermast_, Jan 14 2011

%E More terms from _Amiram Eldar_, Jun 20 2019

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Last modified June 12 07:43 EDT 2021. Contains 344943 sequences. (Running on oeis4.)