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
Keywords
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)
Programs
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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
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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
Comments