A228446 - OEIS

%I #44 Nov 03 2024 06:15:44

%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 a(n) = smallest prime p such that 2*n+1 = p + x*(x+1) for some positive integer x, or -1 if no such prime exists.

%C Based on Sun's conjecture 1.4 in the paper referenced below.

%C The plot shows an ever-widening band of sawtooth shape.  New maxima values will include sequence members larger than the largest prime factor of the original n.  For example when n = 21 with prime factors 3 and 7, and a(10) = 19.

%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, 65-76. (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], 2008-2009.

%e 21 = 19+1*2 where no solution exists using p = 2, 3, 5, 7, 11, 13, 17. So a(10) = 19.

%e 51 = 31+4*5 where no lower odd prime provides a solution. So a(25) = 31.

%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,changed

%O 2,1

%A _Bill McEachen_, Oct 26 2013

%E Entry revised by _N. J. A. Sloane_, Nov 11 2020 (including addition of escape clause).