A261529 Number k such that k^2 + 1 = p*q*r where p,q,r are distinct primes and the sum p+q+r is a perfect square.

17, 37, 91, 235, 683, 1423, 1675, 2879, 8101, 9595, 13711, 18799, 19601, 21295, 25937, 30059, 32111, 36251, 39505, 41071, 49285, 60719, 79441, 90575, 93871, 94799, 103429, 112571, 132085, 136075, 144965, 180001, 180251, 188465, 189679

Offset: 1

Author

Michel Lagneau, Aug 23 2015

Keywords

• nonn

Comments

a(n) is odd. The prime numbers of the sequence are 17, 37, 683, 1423, 2879, 8101, 13711, 30059, 36251, 60719, 93871, 112571, 180001, ...

Examples

			17 is in the sequence because 17^2 + 1 = 2*5*29 and 2 + 5 + 29 = 6^2.
		

Crossrefs

Cf. A002522, A180278.

Programs

Maple

with(numtheory):
for n from 1 to 200000 do:
  y:=factorset(n^2+1):n0:=nops(y):
   if n0=3 and bigomega(n^2+1)=3 and
   sqrt(y[1]+y[2]+y[3])=floor(sqrt(y[1]+y[2]+y[3]))
   then
   printf(`%d, `,n):
   else
   fi:
od:

PARI

isok(n) = my(f = factor(n^2+1)); (#f~ == 3) && (vecmax(f[,2]) == 1) && issquare(vecsum(f[,1])); \\ Michel Marcus, Aug 24 2015