A137996 Numbers which are not the sum of a triangular number and zero or a prime = 1 (mod 4).

2, 4, 7, 9, 12, 22, 24, 25, 31, 46, 48, 70, 75, 80, 85, 87, 93, 121, 126, 135, 148, 162, 169, 186, 205, 211, 213, 216, 220, 222, 246, 255, 315, 331, 357, 375, 396, 420, 432, 441, 468, 573, 588, 615, 690, 717, 735, 738, 750, 796, 879, 924, 1029, 1038, 1080, 1155

Offset: 1

Author

M. F. Hasler, Mar 24 2008

Keywords

• nonn

Comments

Zhi-Wei SUN conjectured that n=216 is the only number not of the form n = p + k(k+1)/2 with p a prime or zero and more precisely that the numbers given in this sequence A137996(1..123) = 2,...,88956 are the only numbers which cannot be written in this form with p=0 or p=1 (mod 4).

Links

• Zhi-Wei SUN, A new conjecture: n=p+x(x+1)/2, Mar 23, 2008.

Crossrefs

Cf. A137997, A132399, A000040, A000217.

Programs

PARI

{ for( n=1,10^8, t=sqrtint(2*n); if( 0 >= p =n-t*(t+1)/2, !p & next, isprime(p) & p%4==1 & next); until( !t--, isprime( p+=t ) & p%4==1 & next(2)); print1(n","))}