A361904 Odd numbers k such that for all even divisors d of k^2+1, d^2+1 is a prime number.

1, 3, 5, 45, 65, 175, 277, 345, 435, 573, 673, 695, 715, 875, 955, 985, 1095, 1255, 1405, 1495, 1515, 1845, 1915, 2035, 2135, 2315, 2375, 2525, 2687, 2805, 2837, 2965, 3035, 3665, 3715, 3725, 4185, 4225, 4265, 4345, 4495, 4635, 4865, 5987, 6195, 6205, 6315

Offset: 1

Author

Michel Lagneau, Mar 28 2023

Keywords

• nonn

Comments

Conjecture: the sequence is infinite.

We observe two subsequences each having specific properties:

(i) a subsequence of numbers divisible by 5 such that a(n)^2+1 contains 4 divisors {1, 2, p, 2*p} including two even divisors 2 and 2*p = a(n)^2+1 where p is prime.

(ii) a subsequence of numbers > 3 not divisible by 5 such that a(n)^2+1 contains eight divisors {1, 2, 5, 10, q, 2*q, 5*q, 10*q} including four even divisors 2, 10, 2*q = (a(n)^2+1)/5 and 10*q = a(n)^2+1, where q is prime.

Examples

			5 is in the sequence because the divisors of 5^2+1 are {1, 2, 13, 26} and 2^2+1 = 5 is prime, 26^2+1 = 677 is prime.
277 is in the sequence because the divisors of 277^2+1 are {1, 2, 5, 10, 7673, 15346, 38365, 76730} and 2^2+1 = 5 is prime, 10^2+1 = 101 is prime, 15346^2+1 = 235499717 is prime, 76730^2+1 = 5887492901 is prime.
		

Links

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

Crossrefs

Cf. A000203, A002522, A193432.

Programs

Maple

q:= k-> andmap(d-> d::odd or isprime(d^2+1), numtheory[divisors](k^2+1)):
select(q, [2*i-1$i=1..4000])[];  # Alois P. Heinz, Mar 30 2023

Mathematica

Lst={};A=Table[Length[Select[Divisors[n^2+1],EvenQ]],{n,1,10000}];B=Array[DivisorSum[#^2+1,1&,And[EvenQ@#,PrimeQ[#^2+1]]&]&,10000];Do[If[A[[m]]==B[[m]]||B[m]!=0||B[m]!=0,AppendTo[Lst,m]],{m,1,10000,2}];Lst

PARI

isok(k) = if (k%2, my(d=select(x->!(x%2), divisors(k^2+1))); for (i=1, #d, if (!isprime(d[i]^2+1), return(0))); return(1)); \\ Michel Marcus, Mar 28 2023

Python

from sympy import divisors, isprime
def ok(n): return n&1 and all(d&1 or isprime(d**2+1) for d in divisors(n**2+1))
print([k for k in range(7000) if ok(k)]) # Michael S. Branicky, Apr 17 2023