A206246 Numbers k such that the greatest prime divisor p of k^2+1 has the property that (p - k)^2 + 1 = p.

1, 3, 7, 13, 21, 31, 43, 91, 111, 183, 211, 241, 273, 381, 421, 553, 601, 651, 703, 1261, 1333, 1561, 1641, 2863, 2971, 3081, 3193, 4291, 4423, 5403, 5551, 6973, 7141, 8011, 8191, 8743, 8931, 11991, 12211, 13341, 13573, 14281, 14521, 15253, 15501, 15751, 16003

Offset

1,2

Comments

For the k > 1 in this sequence, k^2+1 is composite. The corresponding primes p are A002496(n) repeated two times for n > 1 : {2, 5, 5, 17, 17, 37, 37, 101, 101, 197,...}.

Because this sequence is connected with A002496, it is conjectured that the set of this numbers is infinite.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

31 is in the sequence because 31^2 + 1 = 2*13*37 and (37 - 31)^2 + 1 = 37.
43 is in the sequence because 43^2 + 1 = 2*5*5*37 and (37 - 43)^2 + 1 = 37.

Maple

with(numtheory):for n from 1 to 20000 do:x:=n^2+1:y:=factorset(x):n1:=nops(y):p:=y[n1]:q:=(p-n)^2+1:if q=p then printf(`%d, `, n): else fi:od:
# Alternative:
N:= 20000: # for terms <= N
R:= {1, 3}:
for m from 2 by 2 while m^2-m+1 <= N do
   if isprime(m^2+1) then
     R:= select(`<=`, {m^2+m+1, m^2-m+1}, N) union R;
   fi
od:
sort(convert(R, list)); # Robert Israel, Feb 09 2026

Mathematica

pn2pQ[n_]:=Module[{p=FactorInteger[n^2+1][[-1, 1]]}, (p-n)^2+1==p]; Select[ Range[20000], pn2pQ] (* Harvey P. Dale, Nov 20 2019 *)

Crossrefs

Cf. A002496, A134406.

Sequence in context: A353887 A063541 A393251 * A171965 A011898 A098577

Adjacent sequences: A206243 A206244 A206245 * A206247 A206248 A206249

Keyword

nonn

Author

Michel Lagneau, Feb 05 2012

Status

approved