A212643 Let b(n) and c(n) be the total numbers of distinct prime signatures and second signatures, respectively, represented among divisors of A181800(n) (first integers of each second signature; cf. A212172). b(n) mod c(n) = a(n).

0, 1, 1, 1, 1, 0, 1, 4, 1, 5, 4, 1, 6, 5, 1, 7, 6, 2, 1, 8, 5, 7, 2, 1, 9, 6, 8, 2, 1, 10, 7, 1, 9, 2, 6, 1, 11, 8, 0, 10, 2, 7, 1, 12, 9, 18, 0, 11, 2, 8, 15, 1, 13, 10, 22, 0, 7, 14, 12, 2, 9, 20, 1, 14, 11, 26, 7, 8, 18, 13, 2, 10, 25, 1, 15, 15, 12, 30, 9

Offset

1,8

Comments

Significance of the sequence: 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. For all integers n with second signature {S}, A212180(n) = k and A085082(n) is congruent to j modulo k; see examples.

Note: b(n) = A212642(n); c(n) = A212644(n).

Links

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

Formula

a(n) = A212642(n)-A212644(n), reduced modulo A212644(n).

Example

4 is the smallest integer with second signature {2}, and its divisors represent 3 distinct prime signatures and 2 distinct second signatures. 1 = 3 mod 2. Since 4 = A181800(2), a(2) = 1. For all integers m with second signature {2}, A085082(m) is congruent to 1 modulo 2.
10800 is the smallest integer with second signature {4,3,2}, and its divisors represent 28 distinct prime signatures and 14 distinct second signatures. 0 = 28 mod 14.  Since 10800 = A181800(39), a(39) = 0. For all integers m with second signature {4,3,2}, A085082(m) is congruent to 0 modulo 14.

Crossrefs

Cf. A085082, A212171, A212172, A212180, A212642, A212644.

Sequence in context: A329215 A214892 A030352 * A104571 A249057 A105721

Adjacent sequences: A212640 A212641 A212642 * A212644 A212645 A212646

Keyword

nonn,look

Author

Matthew Vandermast, Jun 05 2012

Status

approved