A253596 Numbers k such that A002313(m) is the greatest prime divisor of k^2 + 1 and A002313(m+1) is the greatest prime divisor of (k+1)^2 + 1 for some m.

1, 7, 31, 293, 1936, 2244, 4158, 5744, 11573, 25242, 285202, 339354

Offset

1,2

Comments

A002313 contains the primes congruent to 1 or 2 (mod 4).

The corresponding indices m in A002313 are 1, 2, 6, 13, 69, 65, 322, 199, 130, 46, 1471, 866, ...

The corresponding primes A002313(m) are 2, 5, 37, 101, 809, 761, 4877, 2777, 1709, 509, 26821, 14957, ...

Links

Table of n, a(n) for n=1..12.

Example

31 is in the sequence because 31^2 + 1 = 2*13*37 and 32^2 + 1 = 5*5*41 with the property that 37 = A002313(6) and 41 = A002313(7).

Maple

with(numtheory): nn:=500000:print(1):
for n from 1 to nn do:
   p:=n^2+1:x:=factorset(p):n0:=nops(x):p1:=x[n0]:
   q:=(n+1)^2+1:y:=factorset(q):n1:=nops(y):p2:=y[n1]:ii:=0:
     for j from 2 by 2 to 1000 while(ii=0) do:
      pp:=p1+j:
      if type(pp, prime)=true and irem(pp, 4)=1
      then
      p3:=pp:ii:=1:
      else
      fi:
    od:
    if p3=p2
    then
    print(n):
     else
     fi:
    od:

Mathematica

lst={}; Do[If[Mod[Prime[i], 4]==1||Mod[Prime[i], 4]==2, AppendTo[lst, Prime[i]]], {i, 1, 1000}]; Do[Do[If[FactorInteger[n^2+1][[-1]][[1]]==Part[lst, j]&&FactorInteger[(n+1)^2+1][[-1]][[1]]==Part[lst, j+1], Print[n]], {n, 1, 20000}], {j, 1, 999}]

Crossrefs

Cf. A002313, A014442.

Sequence in context: A143564 A352411 A344787 * A298958 A153028 A276667

Adjacent sequences: A253593 A253594 A253595 * A253597 A253598 A253599

Keyword

nonn,more

Author

Michel Lagneau, Jan 05 2015

Status

approved