This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319036 #13 Dec 08 2018 03:44:53 %S A319036 0,6,153,66,0,3916,0,1770,2556,327645,0,1540,0,893862621,8199225, %T A319036 17766,0,76636,0,12720,662976,2096128,0,10296,3357936, %U A319036 416798777159765703,6221628,3611328,0,1734453,0,303810,111576864636,1420010137134674578503,18051523357140153 %N A319036 a(n) is the smallest triangular number T(k) such that both it and its successor T(k+1) have exactly 2n divisors, or 0 if no such pair of consecutive triangular numbers exists. %C A319036 The only primes p for which a(p) > 0 are those for which both 2*3^(p-1) - 1 and 2*3^(p-1) + 1 are prime: 2, 3, and any other primes p such that p-1 appears both in A003307 and A003306. (If such a prime p > 3 exists, then p exceeds 1360105.) %C A319036 Conjecture: The only primes p for which a(p) > 0 are 2 and 3. %e A319036 For n=1, the only triangular number with exactly 2*1 = 2 divisors is T(2) = 2*(2+1)/2 = 3 (the only triangular number that is prime); thus, exists no pair of consecutive triangular numbers having exactly 2 divisors, so a(1)=0. %e A319036 a(2) is 6 because T(3) = 3*(3+1)/2 = 6 and T(4) = 4*(4+1)/2 = 10 are the first two consecutive triangular numbers having exactly 2*2 = 4 divisors. %Y A319036 Cf. A000005, A000217, A063440, A081978, A319035. %Y A319036 Cf. A003306, A003307. %K A319036 nonn %O A319036 1,2 %A A319036 _Jon E. Schoenfield_, Dec 05 2018