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

%I #26 Mar 18 2024 15:21:07

%S 1,3,5,45,65,175,277,345,435,573,673,695,715,875,955,985,1095,1255,

%T 1405,1495,1515,1845,1915,2035,2135,2315,2375,2525,2687,2805,2837,

%U 2965,3035,3665,3715,3725,4185,4225,4265,4345,4495,4635,4865,5987,6195,6205,6315

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

%C Conjecture: the sequence is infinite.

%C We observe two subsequences each having specific properties:

%C (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.

%C (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.

%H Robert Israel, <a href="/A361904/b361904.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e 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.

%p q:= k-> andmap(d-> d::odd or isprime(d^2+1), numtheory[divisors](k^2+1)):

%p select(q, [2*i-1$i=1..4000])[]; # _Alois P. Heinz_, Mar 30 2023

%t 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

%o (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

%o (Python)

%o from sympy import divisors, isprime

%o def ok(n): return n&1 and all(d&1 or isprime(d**2+1) for d in divisors(n**2+1))

%o print([k for k in range(7000) if ok(k)]) # _Michael S. Branicky_, Apr 17 2023

%Y Cf. A000203, A002522, A193432.

%K nonn

%O 1,2

%A _Michel Lagneau_, Mar 28 2023