This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A045768 #75 May 24 2025 02:05:58
%S A045768 1,20,104,464,650,1952,130304,522752,8382464,134193152,549754241024,
%T A045768 8796086730752,140737463189504,144115187270549504
%N A045768 Numbers k such that sigma(k) == 2 (mod k).
%C A045768 Equivalently, Chowla function of k is congruent to 1 (mod k).
%C A045768 If p=2^i-3 is prime, then 2^(i-1)*p is a term of the sequence. 650 is in the sequence, but is not of this form.
%C A045768 Terms k from a(2) to a(14) satisfy sigma(k) = 2*k + 2, implying that sigma(k) == 0 (mod k+1). It is not known if this holds in general, for there might be solutions of sigma(k)=3k+2 or 4k+2 or ... (Comments from _Jud McCranie_ and _Dean Hickerson_, updated by _Jon E. Schoenfield_, Sep 25 2021 and by _Max Alekseyev_, May 23 2025).
%C A045768 k | sigma(k) produces the multiperfect numbers (A007691). It is an open question whether k | sigma(k) - 1 iff k is a prime or 1. It is not known if there exist solutions to sigma(k) = 2k+1.
%C A045768 Sequence also gives the nonprime solutions to sigma(k) == 0 (mod k+1), k > 1. - _Benoit Cloitre_, Feb 05 2002
%C A045768 Sequence seems to give nonprime k such that the numerator of the sum of the reciprocals of the divisors of k equals k+1 (nonprime k such that A017665(k)=k+1). - _Benoit Cloitre_, Apr 04 2002
%C A045768 For k > 1, composite numbers k such that A108775(k) = floor(sigma(k)/k) = sigma(k) mod k = A054024(k). Complement of primes (A000040) with respect to A230606. There are no numbers k > 2 such that sigma(x) = k*(x+1) has a solution. - _Jaroslav Krizek_, Dec 05 2013
%D A045768 R. K. Guy, Unsolved Problems in Number Theory, B2.
%H A045768 Amitabha Tripathi, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/48-2/Tripathi.pdf">A note on products of primes that differ by a fixed integer</a>, Fibonacci Quart. 48 (2010), no. 2, 144-149.
%e A045768 sigma(650) = 1302 == 2 (mod 650).
%t A045768 Do[If[Mod[DivisorSigma[1, n]-2, n]==0, Print[n]], {n, 1, 10^8}]
%t A045768 Join[{1}, Select[Range[8000000], Mod[DivisorSigma[1, #], #]==2 &]] (* _Vincenzo Librandi_, Mar 11 2014 *)
%o A045768 (PARI) is(n)=sigma(n)%n==2 || n==1 \\ _Charles R Greathouse IV_, Mar 09 2014
%Y A045768 Numbers k such that A054013(k)=1.
%Y A045768 Cf. A181597, A050414, A050415, A054024, A045769, A088834, A045770, A076496.
%K A045768 nonn
%O A045768 1,2
%A A045768 _Dan Hoey_
%E A045768 More terms from _Jud McCranie_, Dec 22 1999.
%E A045768 a(11) from _Donovan Johnson_, Mar 01 2012
%E A045768 a(12) from _Giovanni Resta_, Apr 02 2014
%E A045768 a(13) from _Jud McCranie_, Jun 02 2019
%E A045768 Edited and a(14) from _Jon E. Schoenfield_ confirmed by _Max Alekseyev_, May 23 2025