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

1,2

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

Links

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

Example

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

Mathematica

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

Crossrefs

Cf. A000040, A000217, A046903.

Sequence in context: A163246 A014738 A182677 * A282853 A341046 A162940

Adjacent sequences: A076765 A076766 A076767 * A076769 A076770 A076771

Keyword

nonn

Author

Jason Earls, Nov 14 2002

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

Status

approved