A076768 Positive integers not expressible as the sum of a prime and a triangular number.

1, 36, 105, 171, 210, 216, 325, 351, 406, 528, 561, 630, 741, 780, 990, 1081, 1176, 1275, 1596, 1711, 1830, 1953, 2016, 2145, 2346, 2628, 2775, 3003, 3081, 3240, 3321, 3655, 3741, 3916, 4278, 4371, 4465, 4560, 4851, 5253, 5460, 5565, 5886, 6105, 6216, 6786, 7021, 7140, 7503, 7626, 7750, 7875, 8256, 8515, 8911, 9045, 9591, 9870

Offset: 1

Author

Jason Earls, Nov 14 2002

Keywords

• nonn

Comments

It appears that 1,2,3,8 are the only positive integers that cannot be partitioned as the sum of a semiprime and a triangular number. Here triangular numbers include t(0)=0 and t(1)=1. - Jonathan Vos Post and Ray Chandler, Nov 28 2004

This sequence contains 216 (and possibly other nontriangular numbers) together with an infinite number of triangular numbers. The indices of the triangular numbers are in A138666. This is related to the Sun's conjecture (see A132399) that every number except 216 is the sum of a triangular number and a prime or 0. - T. D. Noe, Mar 26 2008

Examples

			a(2) = 36 is an element of this sequence because 36 cannot be written as a sum of one of the primes <= 36 {2,3,5,7,11,13,17,19,23,29,31} and one of the triangular numbers <= 36 {1,3,6,10,15,21,28,36}. - corrected (added 28) by _Gionata Neri_, May 02 2015
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1001

Crossrefs

Cf. A000040, A000217, A046903.

Programs

Mathematica

Complement[Range[9871],Total/@Tuples[{Prime[Range[1220]],Accumulate[ Range[ 0,140]]}]] (* Harvey P. Dale, Jul 30 2019 *)

Extensions

Added the terms 6786 through 9870 and conjecture that there are no further terms - Jonathan Vos Post and Ray Chandler, Nov 28 2004

Added "positive" to the name - Alex Ratushnyak, Apr 04 2013