A212642 a(n) = number of distinct prime signatures represented among divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

1, 3, 4, 5, 6, 6, 7, 9, 8, 12, 10, 9, 15, 14, 10, 18, 18, 10, 11, 21, 15, 22, 16, 12, 24, 20, 26, 22, 13, 27, 25, 19, 30, 28, 21, 14, 30, 30, 28, 34, 34, 27, 15, 33, 35, 37, 20, 38, 40, 33, 31, 16, 36, 40, 46, 15, 28, 30, 42, 46, 39, 43, 17, 39, 45, 55, 25, 35

Offset: 1

Author

Matthew Vandermast, Jun 05 2012

Keywords

• nonn

Comments

Also, number of divisors of A181800 that are members of A025487.

Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures. Let n be any integer with second signature {S}. Then A212180(n) = k and A085082(n) is congruent to j modulo k. Cf. A212643, A212644.

Examples

			The divisors of 36 represent a total of 6 distinct prime signatures (cf. A085082), as can be seen from the positive exponents, if any, in the canonical prime factorization of each divisor:
{ }: 1 (multiset of positive exponents is the empty multiset)
{1}: 2 (2^1), 3 (3^1)
{1,1}: 6 (2^1*3^1)
{2}: 4 (2^2), 9 (3^2),
{2,1}: 12 (2^2*3^1), 18 (2^1*3^2)
{2,2}: 36 (2^2*3^2)
Since 36 = A181800(6), a(6) = 6.
		

Links

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

Crossrefs

Cf. A181800, A085082, A212171, A212172, A212643, A212644.

Formula

a(n) = A085082(A181800(n)).