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A323747 Smallest triangular number whose number of divisors is the n-th triangular number, or 0 if no such number exists.

%I A323747 #14 Jan 29 2021 02:12:28
%S A323747 1,0,28,496,1631432881,0,8256,2016,41616,0,169878528,2717872128,0
%N A323747 Smallest triangular number whose number of divisors is the n-th triangular number, or 0 if no such number exists.
%C A323747 Additional known terms include a(15)=270480, a(16)=77309214720, a(19)=117261433825538425475625, a(20)=7874496, a(22)=0, a(23)=316659361382400, a(24)=100472400, a(25)=0, a(27)=18951806016, a(28)=35184372088827805696000000, a(31)=20752587086144471040, a(32)=3877678080.
%C A323747 It is known (see the comments and links at A081978) that a(n)=0 for every n such that n*(n+1)/2 is an odd composite not divisible by 3; this includes n = 10, 13, 22, 25, ..., i.e., all n such that n mod 12 is 1 or 10.
%C A323747 Conjectures:
%C A323747 1. a(n) > 0 for every n such that n*(n+1)/2 is even.
%C A323747 2. a(n) = 0 for every n such that n*(n+1)/2 is odd except n = 1, 5, and 9 (whose corresponding values of n*(n+1)/2 are 1, 15, and 45, respectively). Can this be proved for any of the values of n in {14, 17, 18, 21, 26, 29, 30}?
%e A323747 a(1) = 1 because 1 is the only triangular number having A000217(1)=1 divisors.
%e A323747 a(2) = 0 because no triangular number has A000217(2)=3 divisors. (Each number with 3 divisors is the square of a prime, and no such number can be of the form k*(k+1)/2.)
%e A323747 a(3) = 28 because 28 = 7*(7+1)/2 = 2^2 * 7 is the smallest triangular number with A000217(3)=6 divisors.
%e A323747 a(5) = 1631432881 = 13^4 * 239^2 is the only triangular with A000217(5)=15 divisors.
%Y A323747 Cf. A000217, A081978, A116541, A242585, A292989.
%K A323747 nonn,more
%O A323747 1,3
%A A323747 _Jon E. Schoenfield_, May 25 2019
%E A323747 a(6)-a(13) and updated comments from _Jon E. Schoenfield_, Jan 29 2021