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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A003766 A278728 A221689 * A185211 A046182 A092122

Adjacent sequences:  A319033 A319034 A319035 * A319037 A319038 A319039

KEYWORD

nonn

AUTHOR

Jon E. Schoenfield, Dec 05 2018

STATUS

approved

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Last modified August 13 22:55 EDT 2020. Contains 336473 sequences. (Running on oeis4.)