A131651 Positive integers obtained as the difference of two triangular numbers in exactly 4 ways.

15, 21, 27, 30, 33, 35, 39, 42, 51, 54, 55, 57, 60, 65, 66, 69, 70, 77, 78, 84, 85, 87, 91, 93, 95, 102, 108, 110, 111, 114, 115, 119, 120, 123, 125, 129, 130, 132, 133, 138, 140, 141, 143, 145, 154, 155, 156, 159, 161, 168, 170, 174, 177, 182, 183, 185, 186, 187

Offset: 1

Author

John W. Layman, Sep 10 2007

Keywords

• nonn

Comments

It appears that terms of the sequence are all given by a power of 2 times the cube of an odd prime or a power of 2 times a product of two distinct odd primes. (This has been verified for a(n) <= 10000.)

Apparently the integers that have exactly 4 odd divisors. (Verified for a(n) <= 187.) - Philippe Beaudoin, Oct 24 2013

Also numbers that can be expressed as the sum of k > 1 consecutive positive integers in exactly 3 ways; e.g., 7+8 = 15, 4+5+6 = 15 and 1+2+3+4+5 = 15. - Julie Jones, Aug 13 2018

Examples

			15 is in the sequence because 15 = 15 - 0 = 21 - 6 = 36 - 21 = 120 - 105, where all operands are triangular, and in no other way.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001227, A038550, A072502, A285800.

Programs

PARI

isok(n) = {v = vector(n, i, i*(i+1)/2); nb = 0; for (i=1, n, if (ispolygonal(i*(i+1)/2 - n, 3), nb++; if (nb > 4, return (0)););); nb == 4;} \\ Michel Marcus, Jan 14 2014