A212644 If an integer's second signature (cf. A212172) is the n-th to appear among positive integers, a(n) = number of distinct second signatures represented among its divisors.

1, 2, 3, 4, 5, 3, 6, 5, 7, 7, 6, 8, 9, 9, 9, 11, 12, 4, 10, 13, 10, 15, 7, 11, 15, 14, 18, 10, 12, 17, 18, 9, 21, 13, 15, 13, 19, 22, 14, 24, 16, 20, 14, 21, 26, 19, 10, 27, 19, 25, 16, 15, 23, 30, 24, 5, 21, 16, 30, 22, 30, 23, 16, 25, 34, 29, 9, 27, 22, 33

Offset

1,2

Comments

Also, number of divisors of A181800(n) that are members of A181800.

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 (cf. A212642). Let m be any integer with second signature {S}. Then A212180(m) = k and A085082(m) is congruent to j modulo k. If {S} is the second signature of A181800(n), then A085082(m) is congruent to A212643(n) modulo a(n).

Links

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

Formula

a(n) = A212180(A181800(n)).

Example

The divisors of 72 represent 5 distinct second signatures (cf. A212172), as can be seen from the exponents >=2, if any, in the canonical prime factorization of each divisor:
{ }: 1, 2 (prime), 3 (prime), 6 (2*3)
{2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
{3}: 8 (2^3), 24 (2^3*3)
{2,2}: 36 (2^2*3^2)
{3,2}: 72 (2^3*3^2)
Since 72 = A181800(8), a(8) = 5.

Crossrefs

Cf. A181800, A085082, A212172, A212176, A212642, A212643.

Sequence in context: A345018 A309079 A194551 * A351285 A341673 A278057

Adjacent sequences: A212641 A212642 A212643 * A212645 A212646 A212647

Keyword

nonn

Author

Matthew Vandermast, Jun 07 2012

Extensions

Data corrected by Amiram Eldar, Jul 14 2019

Status

approved