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%I #26 Mar 30 2020 04:20:45
%S 3,8,12,144,1020,8040,13860,34840,729180,1728240,3232060,17576520,
%T 39279240,85184880,117649980,778689840,884737920,1225045140,
%U 1771563420,3723878100,3869896140,4574299320,7762395960,12487172640,14348911860,14886940920,21484957560,24137574780
%N Numbers k such that k^2 + 1 = p*q, p and q prime with p == q (mod k).
%C q - p = k with k = 3, 8, 144.
%C The next terms with q - p = k are F(432) = 85738...5984 and F(570) where F(n) is the n-th Fibonacci number. All such terms are in A001906; the next such term, if one exists, has more than 25000 decimal digits. - _Charles R Greathouse IV_, Jan 21 2011
%H Jinyuan Wang, <a href="/A180507/b180507.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3) = 12 because 12^2 + 1 = 5*29 and 29 - 5 = 2*12;
%e a(8) = 34840 because 34840^2 + 1 = 4289 * 283009 and 283009 - 4289 = 278720 = 8*34840.
%p with(numtheory):for k from 1 to 40000 do: x:=k^2+1:y:=factorset(x):yy:=bigomega(x):if
%p yy=2 and irem(y[2],k) =y[1] then printf(`%d, `,k):else fi:od:
%o (PARI) w(m, r) = Vec(x*(1-x)/(1-(m^2+2)*x+x^2) + O(x^r));
%o isok(s, t) = isprime(s) && isprime(s+t);
%o lista(nn) = {my(g, k, m=1, r, u=w(1, nn), v=List([])); for(i=2, r=#u, g=k=(u[i]+sqrtint(5*u[i]^2-4))/2; if(isok(u[i], k), listput(v, k))); while(r>2, u=w(m++, r); for(i=2, #u, k=(m*u[i]+sqrtint((m^2+4)*u[i]^2-4))/2; if(k<g, if(isok(u[i], m*k), listput(v, k)), r=i; break))); Set(v); } \\ _Jinyuan Wang_, Mar 29 2020
%Y Subset of A085722.
%Y Cf. A001906, A002496, A005574, A027862, A134406, A134407.
%K nonn
%O 1,1
%A _Michel Lagneau_, Jan 20 2011
%E More terms from _Charles R Greathouse IV_, Jan 24 2011
%E Missing terms inserted and more terms from _Jinyuan Wang_, Mar 30 2020
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