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%I #69 Sep 26 2021 01:53:02
%S 1,20,104,464,650,1952,130304,522752,8382464,134193152,549754241024,
%T 8796086730752,140737463189504
%N Numbers k such that sigma(k) == 2 (mod k).
%C Equivalently, Chowla function of k is congruent to 1 (mod k).
%C 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 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).
%C 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 Sequence also gives the nonprime solutions to sigma(k) == 0 (mod k+1), k > 1. - _Benoit Cloitre_, Feb 05 2002
%C 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 a(12) > 10^12. - _Donovan Johnson_, Mar 01 2012
%C 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
%C a(13) > 10^13. - _Giovanni Resta_, Apr 02 2014
%C a(14) <= 144115187270549504 = 2^28*(2^29 - 3). - _Jon E. Schoenfield_, Jun 02 2019
%D R. K. Guy, Unsolved Problems in Number Theory, B2.
%H Amitabha Tripathi, <a href="http://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 sigma(650) = 1302 == 2 (mod 650).
%t Do[If[Mod[DivisorSigma[1, n]-2, n]==0, Print[n]], {n, 1, 10^8}]
%t Join[{1}, Select[Range[8000000], Mod[DivisorSigma[1, #], #]==2 &]] (* _Vincenzo Librandi_, Mar 11 2014 *)
%o (PARI) is(n)=sigma(n)%n==2 || n==1 \\ _Charles R Greathouse IV_, Mar 09 2014
%Y Numbers k such that A054013(k)=1.
%Y Cf. A181597, A050414, A050415, A054024, A045769, A088834, A045770, A076496.
%K nonn
%O 1,2
%A _Dan Hoey_
%E More terms from _Jud McCranie_, Dec 22 1999. He says there are no other terms < 4290000000.
%E a(11) from _Donovan Johnson_, Mar 01 2012
%E a(12) from _Giovanni Resta_, Apr 02 2014
%E a(13) from _Jud McCranie_, Jun 02 2019
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