OFFSET
1,2
COMMENTS
Equivalently, Chowla function of k is congruent to 1 (mod k).
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.
For 1 < k <= 140737463189504 this sequence has the property that if sigma(k) == 2 (mod k) then 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).
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.
Sequence also gives the nonprime solutions to sigma(k) == 0 (mod k+1), k > 1. - Benoit Cloitre, Feb 05 2002
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
a(12) > 10^12. - Donovan Johnson, Mar 01 2012
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(13) > 10^13. - Giovanni Resta, Apr 02 2014
a(14) <= 144115187270549504 = 2^28*(2^29 - 3). - Jon E. Schoenfield, Jun 02 2019
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
AUTHOR
EXTENSIONS
More terms from Jud McCranie, Dec 22 1999. He says there are no other terms < 4290000000.
a(11) from Donovan Johnson, Mar 01 2012
a(12) from Giovanni Resta, Apr 02 2014
a(13) from Jud McCranie, Jun 02 2019
STATUS
approved