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.

0, 6, 153, 66, 0, 3916, 0, 1770, 2556, 327645, 0, 1540, 0, 893862621, 8199225, 17766, 0, 76636, 0, 12720, 662976, 2096128, 0, 10296, 3357936, 416798777159765703, 6221628, 3611328, 0, 1734453, 0, 303810, 111576864636, 1420010137134674578503, 18051523357140153

Offset

1,2

Comments

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.)

Conjecture: The only primes p for which a(p) > 0 are 2 and 3.

Links

Table of n, a(n) for n=1..35.

Example

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.
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.

Crossrefs

Cf. A000005, A000217, A063440, A081978, A319035.

Cf. A003306, A003307.

Sequence in context: A278728 A221689 A354752 * A185211 A046182 A092122

Adjacent sequences: A319033 A319034 A319035 * A319037 A319038 A319039

Keyword

nonn

Author

Jon E. Schoenfield, Dec 05 2018

Status

approved