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 A045768 Numbers k such that sigma(k) == 2 (mod k). 26
 1, 20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504 (list; graph; refs; listen; history; text; internal format)
 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 member of the sequence. 650 is in the sequence, but is not of this form. For 1 < k <= 134193152 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). 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 gives also 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. - 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, 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 n such that A054013(n)=1. Cf. A181597, A050414, A050415, A054024, A045769, A088834, A045770, A076496. Sequence in context: A241355 A220207 A189950 * A088831 A063785 A181703 Adjacent sequences:  A045765 A045766 A045767 * A045769 A045770 A045771 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

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Last modified April 6 02:43 EDT 2020. Contains 333267 sequences. (Running on oeis4.)