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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 MccCranie 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

Table of n, a(n) for n=1..13.

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

n such that A054013(n)=1.

Cf. 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

Dan Hoey

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 September 23 02:37 EDT 2019. Contains 327327 sequences. (Running on oeis4.)