%I
%S 3,5,3,5,7,3,5,7,19,3,5,7,17,11,3,5,7,19,11,13,3,5,7,31,11,13,37,3,5,
%T 7,23,11,13,29,17,3,5,7,61,11,13,31,17,19,3,5,7,43,11,13,103,17,19,
%U 109,3,5,7,29,11,13,53,17,19,41,23,3,5,7,31,11,13,37
%N For odd n >= 5, lowest prime p such that n = p + x*(x+1) for some x > 0.
%C This is Sun's conjecture 1.4 in the paper listed below.
%C The plot shows an everwidening band of sawtooth shape. New maxima values will include sequence members larger than the largest prime factor of the original n. An example is 19 encountered from n=21=3*7, 19>7.
%C a(A000124(n)) = 3; a(A133263(n)) = 5; a(A167614(n)) = 7.  _Reinhard Zumkeller_, Mar 12 2014
%D Z. W. Sun, On sums of primes and triangular numbers, Journal of Combinatorics and Number Theory 1(2009), no. 1, 6576. (See Conjecture 1.4.)
%H T. D. Noe, <a href="/A228446/b228446.txt">Table of n, a(n) for n = 2..1000</a>
%H Z. W. Sun, <a href="http://arxiv.org/abs/0803.3737">On sums of primes and triangular numbers</a>, arXiv:0803.3737 [math.NT]
%e 21 = 19+1*2 where no solution exists using p = 2, 3, 5, 7, 11, 13, 17.
%e 51 = 31+4*5 where no lower odd prime provides a solution for odd 51.
%t nn = 14; ob = Table[n*(n+1), {n, nn}]; Table[p = Min[Select[n  ob, # > 0 && PrimeQ[#] &]]; p, {n, 5, ob[[1]], 2}] (* _T. D. Noe_, Oct 27 2013 *)
%o (PARI) a(n) = {oddn = 2*n+1; x = oddn; while (! isprime(oddn  x*(x+1)), x); oddn  x*(x+1);} \\ _Michel Marcus_, Oct 27 2013
%o (Haskell)
%o a228446 n = head
%o [q  let m = 2 * n + 1,
%o q < map (m ) $ reverse $ takeWhile (< m) $ tail a002378_list,
%o a010051 q == 1]
%o  _Reinhard Zumkeller_, Mar 12 2014
%Y Cf. A010051, A002378, A000217.
%K easy,nonn
%O 2,1
%A _Bill McEachen_, Oct 26 2013
