A192771 Numbers k such that k^2 + 1 is divisible by precisely five distinct primes where the sum of the largest and the smallest is equal to the sum of the other three.

2153, 2697, 8487, 11293, 12553, 18065, 32247, 43999, 55945, 107607, 134223, 214641, 218783, 366937, 429855, 595471, 620865, 645327, 1330849, 1363977, 1387689, 1532465, 1557535, 1631191, 1716663, 1778711, 2156031, 3166415, 3857215, 4546071

Offset: 1

Author

Michel Lagneau, Jul 09 2011

Keywords

• nonn

Examples

			11293 is in the sequence because 11293^2+1 = 2 * 5 ^ 2 * 29 * 281 * 313 and 313 + 2 = 5 + 29 + 281 = 315.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..144 (terms below 10^9, terms 1..77 from Lukas Naatz)

Crossrefs

Cf. A180278, A192770.

Programs

Maple

isA192771 := proc(n) local p,s1,n2 ; n2 := n^2+1 ; if A001221(n2) = 5 then p := numtheory[factorset](n2) ; s1 := max(op(p)) + min( op(p)) ; evalb( add(k,k=p) = 2*s1 ) ; else false; end if; end proc:
for n from 1 do if isA192771(n) then printf("%d,\n",n); end if; end do: # R. J. Mathar, Jul 11 2011

Mathematica

seqQ[n_] := Module[{p = FactorInteger[n^2 + 1][[;;,1]]}, Length[p] == 5 && p[[1]] + p[[5]] == p[[2]] + p[[3]] + p[[4]]]; Select[Range[10^6], seqQ] (* Amiram Eldar, Jan 15 2020 *)

PARI

for(k=1,5000000,my(f=factor(k^2+1));if(#f[,2]==5,if(f[1,1]+f[5,1]==f[2,1]+f[3,1]+f[4,1],print1(k,", ")))) \\ Hugo Pfoertner, Jan 08 2020

Extensions

a(17) and beyond from Lukas Naatz, Jan 08 2020