A137996 - OEIS

%I #7 Sep 11 2018 08:45:10

%S 2,4,7,9,12,22,24,25,31,46,48,70,75,80,85,87,93,121,126,135,148,162,

%T 169,186,205,211,213,216,220,222,246,255,315,331,357,375,396,420,432,

%U 441,468,573,588,615,690,717,735,738,750,796,879,924,1029,1038,1080,1155

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

%C 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).

%H Zhi-Wei SUN, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;9ebd2e77.0803">A new conjecture: n=p+x(x+1)/2</a>, Mar 23, 2008.

%o (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","))}

%Y Cf. A137997, A132399, A000040, A000217.

%K nonn

%O 1,1

%A _M. F. Hasler_, Mar 24 2008