OFFSET
1,2
COMMENTS
Equivalently, Chowla function (A048050) of k is congruent to 1 (mod k). Same as A054013(k) = 1 for k > 1.
If p=2^i-3 is prime (i is a term of A050414), then 2^(i-1)*p is a term of the sequence. 650 is in the sequence, but is not of this form.
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).
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. In particular, it is not known if there exist solutions to sigma(k) = 2k+1, called quasiperfect numbers.
All known terms k > 1 (which are also terms of A088831) satisfy the congruence sigma(k) == 0 (mod k+1), although it may not hold in general. - Benoit Cloitre, Feb 05 2002; updated by Max Alekseyev, Feb 05 2026
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
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
a(15) > 10^24. - Max Alekseyev, Feb 05 2026
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, B2.
LINKS
Amitabha Tripathi, A note on products of primes that differ by a fixed integer, Fibonacci Quart. 48 (2010), no. 2, 144-149.
EXAMPLE
sigma(650) = 1302 == 2 (mod 650).
MATHEMATICA
Do[If[Mod[DivisorSigma[1, n]-2, n]==0, Print[n]], {n, 1, 10^8}]
Join[{1}, Select[Range[8000000], Mod[DivisorSigma[1, #], #]==2 &]] (* Vincenzo Librandi, Mar 11 2014 *)
PROG
(PARI) is(n)=sigma(n)%n==2 || n==1 \\ Charles R Greathouse IV, Mar 09 2014
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
EXTENSIONS
More terms from Jud McCranie, Dec 22 1999.
a(11) from Donovan Johnson, Mar 01 2012
a(12) from Giovanni Resta, Apr 02 2014
a(13) from Jud McCranie, Jun 02 2019
Edited and a(14) from Jon E. Schoenfield confirmed by Max Alekseyev, May 23 2025
STATUS
approved