cp's OEIS Frontend

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).

%I A212643 #17 Jul 16 2024 14:17:01
%S A212643 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,
%T A212643 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,
%U A212643 1,14,11,26,7,8,18,13,2,10,25,1,15,15,12,30,9
%N 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).
%C A212643 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.
%C A212643 Note: b(n) = A212642(n); c(n) = A212644(n).
%H A212643 Amiram Eldar, <a href="/A212643/b212643.txt">Table of n, a(n) for n = 1..10000</a>
%F A212643 a(n) = A212642(n)-A212644(n), reduced modulo A212644(n).
%e A212643 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.
%e A212643 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.
%Y A212643 Cf. A085082, A212171, A212172, A212180, A212642, A212644.
%K A212643 nonn,look
%O A212643 1,8
%A A212643 _Matthew Vandermast_, Jun 05 2012