A360107 -id:A360107 - cp's OEIS frontend

A360105 Numbers k such that sigma_2(k^2 + 1) == 0 (mod k).

1, 2, 5, 7, 13, 25, 34, 52, 89, 93, 100, 200, 233, 338, 610, 850, 915, 1028, 1352, 1508, 1918, 2105, 3918, 4181, 5540, 6396, 6728, 7250, 9282, 10100, 10132, 10946, 15507, 16609, 17125, 32708, 32776, 37107, 42568, 47770, 58218, 61230, 72125, 74948, 75025, 78608

Offset: 1

Michel Lagneau, Jan 26 2023

nonn

Conjecture: the sequence contains infinitely many Fibonacci numbers (see A360107).

For k < 10^7, we observe only 6 prime numbers in the sequence: {2, 5, 7, 13, 89, 233} including the Fibonacci numbers {2, 5, 13, 89, 233} and the Lucas number {7}.

			7 is in the sequence because the divisors of 7^2+1 = 50 are {1, 2, 5, 10, 25, 50}, and 1^2 + 2^2 + 5^2 + 10^2 + 25^2 + 50^2 = 3255 = 7*465 == 0 (mod 7).

Robert Israel, Table of n, a(n) for n = 1..222

Cf. A000032, A000045, A001157, A067719, A360107.

filter:= k -> NumberTheory:-SumOfDivisors(k^2+1,2) mod k = 0:
select(filter, [$1..10^5]); # Robert Israel, Feb 19 2024

Select[Range[50000], Divisible[DivisorSigma[2, #^2+1], #]&]

isok(k) = sigma(k^2 + 1, 2) % k == 0; \\ Michel Marcus, Jan 26 2023

cp's OEIS Frontend

A360105 Numbers k such that sigma_2(k^2 + 1) == 0 (mod k).

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