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%I #20 Jul 14 2020 10:47:56
%S 2,5,6,7,9,10,11,13,19,21,22,25,27,30,33,37,42,43,51,57,58,61,67,70,
%T 73,75,78,82,85,91,93,102,105,115,121,123,130,133,145,147,157,163,165,
%U 177,187,190,193,205,210,211,213,217,235,253,267,273,277,283,310
%N Sum-prime (SP) numbers: numbers k that are not divisible by 4 such that (a + b)/gcd(k+1, 4) is prime for every factorization k = a*b.
%C The number of SP numbers below 10^n: 5, 31, 123, 532, 2728, 15402, 98294, ...
%H Amiram Eldar, <a href="/A303437/b303437.txt">Table of n, a(n) for n = 1..10000</a>
%H Kenichi Shimizu, <a href="http://ousar.lib.okayama-u.ac.jp/56015">Arithmetic of Positive Integers Having Prime Sums of Complementary Divisors</a>, Math. J. Okayama Univ., Vol. 60 (2018), pp. 155-164.
%H Kenichi Shimizu and Kazuo Goto, <a href="https://repository.lib.tottori-u.ac.jp/4656">A Property of Integers Related to Quadratic Fields</a>, J. Fac. Educ. Tottori Univ. (Nat. Sci.), Vol. 47 (1998) pp. 5-12.
%e 9 is an SP number since 9 = 1 * 9 = 3 * 3 and (1 + 9)/2 = 5 and (3 + 3)/2 = 3 are both primes.
%e 42 is an SP number since 42 = 1 * 42 = 2 * 21 = 3 * 14 = 6 * 7, and 1 + 42 = 43, 2 + 21 = 23, 3 + 14 = 17, and 6 + 7 = 13 are all primes.
%t div[n_]:=Switch[Mod[n,4],0,0,1,2,2,1,3,4]; pQ[n_,div_] := Module[{}, d=Divisors[n]; m=Length[d];mm=If[OddQ[m],(m+1)/2,m/2];dd=Take[d,mm]; ret=True; Do[p=(dd[[k]]+n/dd[[k]])/div; If[!PrimeQ[p],ret=False;Break[]],{k,1,mm}];ret]; spQ[n_]:=Module[{div1=div[n],ret},If[div1==0,ret=False,ret=pQ[n,div1]];ret];s={};Do[If[spQ[n],AppendTo[s,n]],{n,1,100}];s
%o (PARI) isok(n) = {if (n % 4, my (d = divisors(n)); for (i=1, ceil(#d/2), if (! isprime((d[i]+n/d[i])/gcd(n+1, 4)), return(0));); return (1););} \\ _Michel Marcus_, Apr 26 2018
%Y Subsequence of A042968.
%K nonn
%O 1,1
%A _Amiram Eldar_, Apr 24 2018
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