A180507 Numbers k such that k^2 + 1 = p*q, p and q prime with p == q (mod k).

3, 8, 12, 144, 1020, 8040, 13860, 34840, 729180, 1728240, 3232060, 17576520, 39279240, 85184880, 117649980, 778689840, 884737920, 1225045140, 1771563420, 3723878100, 3869896140, 4574299320, 7762395960, 12487172640, 14348911860, 14886940920, 21484957560, 24137574780

Offset: 1

Author

Michel Lagneau, Jan 20 2011

Keywords

• nonn

Comments

q - p = k with k = 3, 8, 144.

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

Examples

			a(3) = 12 because 12^2 + 1 = 5*29 and 29 - 5 = 2*12;
a(8) = 34840 because 34840^2 + 1 = 4289 * 283009 and 283009 - 4289 = 278720 = 8*34840.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..10000

Crossrefs

Subset of A085722.

Cf. A001906, A002496, A005574, A027862, A134406, A134407.

Programs

Maple

with(numtheory):for k from 1 to 40000 do: x:=k^2+1:y:=factorset(x):yy:=bigomega(x):if
  yy=2 and irem(y[2],k) =y[1] then printf(`%d, `,k):else fi:od:

PARI

w(m, r) = Vec(x*(1-x)/(1-(m^2+2)*x+x^2) + O(x^r));
isok(s, t) = isprime(s) && isprime(s+t);
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(kJinyuan Wang, Mar 29 2020

Extensions

More terms from Charles R Greathouse IV, Jan 24 2011

Missing terms inserted and more terms from Jinyuan Wang, Mar 30 2020