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A360105
Numbers k such that sigma_2(k^2 + 1) == 0 (mod k).
2
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jan 26 2023
STATUS
approved