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%I #24 Sep 08 2022 08:46:11
%S 2,7,61,211,216
%N Numbers that are neither triangular nor the sum of a positive triangular number and a prime.
%C Probably finite.
%C First four terms are primes, and are in A065397.
%C If it exists, a(6) > 5*10^6. - _Derek Orr_, Mar 12 2015
%C If it exists, a(6) > 4*10^9. - _Hiroaki Yamanouchi_, Mar 16 2015
%e a(2) = 7 is in the sequence because 7 is not a triangular number, and cannot be written as a sum of the primes < 7 {2, 3, 5} and one of the positive triangular numbers < 7 {1, 3, 6}.
%o (Magma) lst:=[]; r:=21; for n in [0..r*(r+1)/2] do if not IsSquare(8*n+1) then f:=0; k:=1; while k*(k+1)/2 lt n-1 do if IsPrime(n-Truncate(k*(k+1)/2)) then f:=1; break; end if; k+:=1; end while; if IsZero(f) then Append(~lst, n); end if; end if; end for; lst;
%o (PARI) a(n)=k=1;while(n-(t=k*(k+1)/2)>=0,if(isprime(n-t)||n==t,return(k));k++)
%o n=1;while(n<10^5,if(!a(n),print1(n,", "));n++) \\ _Derek Orr_, Mar 12 2015
%Y Cf. A065397, A076768.
%K nonn,fini
%O 1,1
%A _Arkadiusz Wesolowski_, Mar 10 2015
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