A046903 -id:A046903 - cp's OEIS frontend

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

Jason Earls, Nov 14 2002

nonn

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

			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

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

Cf. A000040, A000217, A046903.

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

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

A100570 Numbers that are not the sum of a square and a semiprime.

Original entry on oeis.org

1, 2, 3, 12, 17, 28, 32, 72, 108, 117, 297, 657

Offset: 1

Jonathan Vos Post, Nov 29 2004

No others up to 300000. Computed in collaboration with Ray Chandler. It appears that this sequence is finite, that is, that almost every positive integer is the sum of a semiprime and a square number. There are probably no further exceptions after a(12)=657.
The statement about the finiteness of this sequence (namely, a(n)<=657) is much stronger than the Goldbach binary conjecture. Indeed, a much weaker conjecture, that this sequence contains no perfect squares >1, already implies the Goldbach conjecture. Cf. comment in A241922. - Vladimir Shevelev, May 01 2014
From Daniel Mikhail, Nov 23 2020: (Start)
There are no new terms in this sequence between 658 and 2^28.
Notably, A014090 (numbers that are not the sum of a square and one prime) is a known infinite sequence. (End)

			From _Daniel Mikhail_, Nov 23 2020: (Start)
An integer m is in this set if, for any primes, p and q, there does not exist a natural k, such that m-k^2 = p*q.
Consider m=12 and all k such that k^2 < 12: k is either 0,1,4, or 9.
  12 - 0 = 12 = 2*2*2*3 => not semiprime;
  12 - 1 = 11 => not semiprime;
  12 - 4 = 8 = 2*2*2 => not semiprime;
  12 - 9 = 3 => not semiprime.
Therefore, 12 is a term. (End)

Daniel Mikhail, Brief glance of the search for integer solutions to a(n)-k^2 = semiprime

Cf. A000290, A001358, A046903, A241922, A014090.

lim = 657; Complement[Range[lim],Select[Flatten[Outer[Plus,Select[Range[lim], PrimeOmega[#] == 2 &],Table[i^2, {i, 0, Sqrt[lim]}]]], # <= lim &]] (* Robert Price, Apr 10 2019 *)

An integer is not an element for any integers i, j of the pairwise sum of {A001358(i)} and {A000290(j)}.

A055108 Largest odd number that can be represented in exactly n ways as p+2*i^2 where p is 1 or a prime and i >= 0.

Original entry on oeis.org

5993, 6797, 59117, 48143, 87677, 148397, 147347, 268157, 285863, 361127, 597473, 448667, 542627, 809057, 753257, 944567, 1281473, 1237007, 1417697, 2148827, 1612067, 2419337, 2550137, 2490587, 2571263, 2565893, 2884823, 2931167, 3383837, 3601067, 3756407

Offset: 0

N. J. A. Sloane

These are just the largest numbers presently known - it has not been proved that they are really the largest.

Sean A. Irvine, Table of n, a(n) for n = 0..60
L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
Sean A. Irvine, Java program (github)
Index entries for sequences related to Goldbach conjecture

Cf. A046903, A007697, A016067.

The sequence as given in the Hodge paper is incorrect; corrected and extended by Jud McCranie, Jun 12 2000

a(11)-a(30) from Donovan Johnson, Mar 21 2012

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

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A100570 Numbers that are not the sum of a square and a semiprime.

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A055108 Largest odd number that can be represented in exactly n ways as p+2*i^2 where p is 1 or a prime and i >= 0.

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