A261530 Numbers k such that k^2 + 1 = p*q*r*s where p,q,r,s are distinct primes and the sum p+q+r+s is a perfect square.

173, 187, 477, 565, 965, 1237, 1277, 1437, 1525, 1636, 2452, 2587, 2608, 2653, 2827, 2885, 2971, 3197, 3388, 3412, 3435, 3477, 3848, 3891, 4188, 4237, 4492, 4724, 5333, 5728, 5899, 6272, 7088, 7108, 7421, 8363, 8541, 9379, 9652, 10227, 10872, 11581, 12237

Offset: 1

Author

Michel Lagneau, Aug 24 2015

Keywords

• nonn

Comments

The primes in the sequence are 173, 1237, 1277, 2971, 5333, 8363, 19387, 20773, ...

The corresponding squares p+q+r+s are 121, 289, 441, 289, 529, 9025, 841, 5625, 529, 196, 5476, 3025, ...

Examples

			173 is in the sequence because 173^2 + 1 = 2*5*41*73 and 2 + 5 + 41 + 73 = 11^2.
		

Crossrefs

Cf. A002522, A261529.

Programs

Maple

with(numtheory):
for n from 1 to 20000 do:
  y:=factorset(n^2+1):n0:=nops(y):
   if n0=4 and bigomega(n^2+1)=4 and
   sqrt(y[1]+y[2]+y[3]+y[4])=floor(sqrt(y[1]+y[2]+y[3]+y[4]))
   then
   printf(`%d, `, n):
   else
   fi:
od:

PARI

isok(n) = my(f = factor(n^2+1)); (#f~== 4) && (vecmax(f[,2]) == 1) && issquare(vecsum(f[,1])) ; \\ Michel Marcus, Aug 24 2015