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

1,1

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, ...

Links

Table of n, a(n) for n=1..43.

Example

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

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:

Prog

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

Crossrefs

Cf. A002522, A261529.

Sequence in context: A259017 A097845 A364937 * A246135 A140002 A178652

Adjacent sequences: A261527 A261528 A261529 * A261531 A261532 A261533

Keyword

nonn

Author

Michel Lagneau, Aug 24 2015

Status

approved