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

1,2

Comments

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}.

Links

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

Example

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).

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A106889 A155028 A119839 * A107057 A212319 A161379

Adjacent sequences: A360102 A360103 A360104 * A360106 A360107 A360108

Keyword

nonn

Author

Michel Lagneau, Jan 26 2023

Status

approved