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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.

%I A212644 #14 Jul 14 2019 10:59:52
%S A212644 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,
%T A212644 12,17,18,9,21,13,15,13,19,22,14,24,16,20,14,21,26,19,10,27,19,25,16,
%U A212644 15,23,30,24,5,21,16,30,22,30,23,16,25,34,29,9,27,22,33
%N 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.
%C A212644 Also, number of divisors of A181800(n) that are members of A181800.
%C A212644 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).
%H A212644 Amiram Eldar, <a href="/A212644/b212644.txt">Table of n, a(n) for n = 1..10000</a>
%F A212644 a(n) = A212180(A181800(n)).
%e A212644 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:
%e A212644 { }: 1, 2 (prime), 3 (prime), 6 (2*3)
%e A212644 {2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
%e A212644 {3}: 8 (2^3), 24 (2^3*3)
%e A212644 {2,2}: 36 (2^2*3^2)
%e A212644 {3,2}: 72 (2^3*3^2)
%e A212644 Since 72 = A181800(8), a(8) = 5.
%Y A212644 Cf. A181800, A085082, A212172, A212176, A212642, A212643.
%K A212644 nonn
%O A212644 1,2
%A A212644 _Matthew Vandermast_, Jun 07 2012
%E A212644 Data corrected by _Amiram Eldar_, Jul 14 2019