

A131651


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


8



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
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OFFSET

1,1


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


LINKS

Table of n, a(n) for n=1..58.


EXAMPLE

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.


PROG

(PARI) isok(n) = sumdiv(n, d, d%2) == 4; \\ conjectured, see Beaudoin comment, Michel Marcus, Oct 29 2013
(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


CROSSREFS

Cf. A001227, A038550, A072502.
Sequence in context: A211375 A217078 A138594 * A134642 A072974 A226025
Adjacent sequences: A131648 A131649 A131650 * A131652 A131653 A131654


KEYWORD

nonn


AUTHOR

John W. Layman, Sep 10 2007


STATUS

approved



