A054013 -id:A054013 - cp's OEIS frontend

A045768 Numbers k such that sigma(k) == 2 (mod k).

1, 20, 104, 464, 650, 1952, 130304, 522752, 8382464, 134193152, 549754241024, 8796086730752, 140737463189504, 144115187270549504

Offset: 1

Dan Hoey

nonn

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.
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. 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
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

			sigma(650) = 1302 == 2 (mod 650).

R. K. Guy, Unsolved Problems in Number Theory, B2.

Amitabha Tripathi, A note on products of primes that differ by a fixed integer, Fibonacci Quart. 48 (2010), no. 2, 144-149.

Numbers k such that A054013(k)=1.

Cf. A181597, A050414, A050415, A054024, A045769, A088834, A045770, A076496.

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 *)

is(n)=sigma(n)%n==2 || n==1 \\ Charles R Greathouse IV, Mar 09 2014

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

A054014 Chowla function of n read modulo (the number of divisors of n).

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 2, 0, 3, 0, 3, 0, 1, 0, 4, 0, 2, 0, 3, 2, 1, 0, 3, 2, 3, 0, 3, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 0, 1, 0, 5, 0, 3, 2, 1, 0, 5, 1, 0, 0, 3, 0, 1, 0, 7, 2, 3, 0, 11, 0, 1, 4, 6, 2, 5, 0, 3, 2, 1, 0, 2, 0, 3, 0, 3, 2, 1, 0, 5, 4, 3, 0, 7, 2, 1, 0, 3, 0, 11, 0, 3, 2, 1, 0, 11, 0, 0, 2, 8, 0, 1, 0

Offset: 1

Asher Auel, Jan 17 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A048050, A054013, A054015.

with(numtheory): [seq((sigma(i)-i-1) mod tau(i),i=1..120)];

Array[Mod[DivisorSigma[1, #] - # - 1, DivisorSigma[0, #]] &, 103] (* Michael De Vlieger, Oct 30 2017 *)

A054014(n) = ((sigma(n)-n-1) % numdiv(n)); \\ Antti Karttunen, Oct 30 2017

a(n) = A048050(n) mod A000005(n).

A054015 a(n) is Chowla function of n read modulo (number of proper divisors of n), a(1) = 0 by convention.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 6, 0, 0, 2, 1, 0, 6, 0, 0, 1, 0, 0, 4, 0, 4, 2, 1, 0, 3, 1, 2, 2, 0, 0, 2, 1, 0, 1, 1, 0, 8, 0, 0, 0, 2, 0, 0, 0, 2, 2, 3, 0, 1, 0, 0, 3, 3, 0, 5, 0, 6, 3, 1, 0, 7, 1, 0, 2, 0, 0, 0, 2, 0, 1, 1, 0, 1, 0, 2, 1, 4, 0, 1, 0, 0, 2

Offset: 1

Asher Auel, Jan 17 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A032741, A048050, A054013, A054014.

with(numtheory): [seq((sigma(i)-i-1) mod (tau(i)-1),i=2..120);#i>1

A054015(n) = if(1==n,0,((sigma(n)-n-1) % (numdiv(n)-1))); \\ Antti Karttunen, Oct 20 2017

a(1) = 0; for n > 1, a(n) = A048050(n) mod A032741(n).

Description clarified by Antti Karttunen, Oct 20 2017

A054017 Chowla's function of n modulo n is a square (excluding 0's).

Original entry on oeis.org

14, 20, 39, 40, 46, 55, 80, 94, 100, 104, 117, 130, 155, 158, 183, 190, 200, 203, 291, 292, 295, 299, 320, 323, 334, 416, 430, 446, 464, 475, 488, 530, 539, 549, 567, 579, 583, 638, 650, 695, 718, 799, 873, 878, 890, 943, 955, 959, 964, 979, 1030, 1118

Offset: 1

Asher Auel, Jan 19 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048050, A054013, A054018.

aQ[n_] := (c = DivisorSigma[1, n] - n - 1) > 0 && IntegerQ @ Sqrt @ Mod[c, n]; Select[Range[1000], aQ] (* Amiram Eldar, Aug 28 2019 *)

A054018 Squares mentioned in A054017.

Original entry on oeis.org

9, 1, 16, 9, 25, 16, 25, 49, 16, 1, 64, 121, 36, 81, 64, 169, 64, 36, 100, 225, 64, 36, 121, 36, 169, 49, 361, 225, 1, 144, 441, 441, 144, 256, 400, 196, 64, 441, 1, 144, 361, 64, 400, 441, 729, 64, 196, 144, 729, 100, 841, 729, 25, 400, 256, 1225, 100, 729, 1225

Offset: 1

Asher Auel, Jan 19 2000

nonn

If Chowla's function of n read (modulo n) is a nonzero square, print this square.

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048050, A054013, A054017, A054019.

chowla[n_] := DivisorSigma[1, n] - n - 1; aQ[n_] := (c = chowla[n]) > 0 && IntegerQ @ Sqrt @ Mod[c, n]; Mod[chowla[#], #] & /@ Select[Range[1000], aQ] (* Amiram Eldar, Aug 28 2019 *)

A054019 Square roots of A054018.

Original entry on oeis.org

3, 1, 4, 3, 5, 4, 5, 7, 4, 1, 8, 11, 6, 9, 8, 13, 8, 6, 10, 15, 8, 6, 11, 6, 13, 7, 19, 15, 1, 12, 21, 21, 12, 16, 20, 14, 8, 21, 1, 12, 19, 8, 20, 21, 27, 8, 14, 12, 27, 10, 29, 27, 5, 20, 16, 35, 10, 27, 35, 31, 30, 29, 3, 12, 28, 5, 1, 35, 26, 10, 20, 37, 12, 33, 18, 43, 43, 45, 22

Offset: 1

Asher Auel, Jan 19 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A048050, A054013, A054017, A054018.

chowla[n_] := DivisorSigma[1, n] - n - 1; aQ[n_] := (c = chowla[n]) > 0 && IntegerQ@Sqrt@Mod[c, n]; Sqrt @ Mod[chowla[#], #] & /@ Select[Range[1000], aQ] (* Amiram Eldar, Aug 28 2019 *)

cp's OEIS Frontend

A045768 Numbers k such that sigma(k) == 2 (mod k).

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A054014 Chowla function of n read modulo (the number of divisors of n).

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A054015 a(n) is Chowla function of n read modulo (number of proper divisors of n), a(1) = 0 by convention.

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A054017 Chowla's function of n modulo n is a square (excluding 0's).

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A054018 Squares mentioned in A054017.

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A054019 Square roots of A054018.

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