A054024 -id:A054024 - cp's OEIS frontend

A076496 Numbers k such that sigma(k) == 12 (mod k).

1, 6, 11, 24, 30, 42, 54, 66, 78, 102, 114, 121, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 780, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338

Offset: 1

Labos Elemer, Oct 21 2002

nonn

			6*p is a solution if p > 3 is prime, since sigma(6*p) = 1 + 2 + 3 + 6 + p + 2*p + 3*p + 6*p = 12*(p+1) = 2*6*p + 12 = 2*k + 12. These are "regular" solutions. Also k = 121, 304 are "singular" solutions. See other remainders in cross-references.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000, corrected by _Sidney Cadot_, Feb 05 2023

Cf. A054024, A045768, A045769, A045770, A076495, A088834.

Cf. A141545 (a subsequence).

Select[Range[2000], Mod[DivisorSigma[1, #] - 12, #] == 0 &] (* Vincenzo Librandi, Mar 11 2014, corrected by Amiram Eldar, Jan 04 2023 *)

isok(k) = Mod(sigma(k), k) == 12; \\ Michel Marcus, Jan 04 2023

Initial term 1 added by Vincenzo Librandi, Mar 11 2014

Terms 6 and 11 inserted by Michel Marcus, Jan 04 2023

A045769 Numbers k such that sigma(k) == 4 (mod k).

Original entry on oeis.org

1, 3, 9, 12, 70, 88, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088, 9223372026117357568

Offset: 1

Dan Hoey

Every number of the form 2^(j-1)*(2^j - 5), where 2^j - 5 is prime, is a term. See A059608. - Jon E. Schoenfield, Jun 02 2019

Contains subsequence A088832.

Cf. A054024, A045768, A059608, A088834, A045770, A076496.

isok(k) = Mod(sigma(k), k) == 4; \\ Michel Marcus, Jan 04 2023

a(13) from Harvey P. Dale, Mar 20 2011
Initial term 1 inserted and a(14)-a(16) from Donovan Johnson, Mar 01 2012
Term 3 inserted by Michel Marcus, Jan 04 2023
a(18) from Jon E. Schoenfield confirmed, and a(17), a(19) added by Max Alekseyev, Jun 08 2025

A088834 Numbers k such that sigma(k) == 6 (mod k).

Original entry on oeis.org

1, 5, 6, 25, 180, 8925, 32445, 442365

Offset: 1

Labos Elemer, Oct 29 2003

For each integer j in A059609, 2^(j-1)*(2^j - 7) is in the sequence. E.g., for j = A059609(1) = 39 we get 151115727449904501489664. - M. F. Hasler and Farideh Firoozbakht, Dec 03 2013
No more terms to 10^10. - Charles R Greathouse IV, Dec 05 2013
a(9) > 10^13. - Giovanni Resta, Apr 02 2014
a(9) > 1.5*10^14. - Jud McCranie, Jun 02 2019
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			Sigma(25) = 31 = 1*25 + 6, so 31 mod 25 = 6.

Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13 (2010), 18 pp. Article ID 10.3.1.

Cf. A054024, A045768, A045769, A045770, A077374, A076496, A088012.
Cf. A087167 (a subsequence).
Cf. A059609.

Select[Range[1000000], Mod[DivisorSigma[1, #] - 6, #] == 0 &] (* T. D. Noe, Dec 03 2013 *)

isok(n) = Mod(sigma(n), n) == 6; \\ Michel Marcus, Jan 03 2023

Terms corrected by Charles R Greathouse IV and Farideh Firoozbakht, Dec 03 2013

A108775 a(n) = floor(sigma(n)/n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Franz Vrabec, Jun 27 2005

nonn

The sequence is unbounded. - Vrabec
First occurrence of k: 1,6,120,27720,..., which is A023199. - Robert G. Wilson v, Jun 28 2005
a(n) > 1 if n is perfect or abundant. a(n) = 2 if n is perfect or primitive abundant (see A091191). - Alonso del Arte, Feb 06 2012

			a(6) = 2 because sigma(6)/6 = (1 + 2 + 3 + 6)/6 = 2.

W. Sierpinski, Elementary Theory of Numbers, 1987, p. 174 ff.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000203, A054024.

Cf. A017665, A017666.

a108775 n = div (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

Table[ Floor[ DivisorSigma[1, n]/n], {n, 105}] (* Robert G. Wilson v, Jun 28 2005 *)

a(n) = sigma(n)\n; \\ Michel Marcus, Sep 18 2015

a(n) = floor(A017665(n)/A017666(n)). - Michel Marcus, Sep 18 2015

More terms from Robert G. Wilson v, Jun 28 2005

A242480 a(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n.

Original entry on oeis.org

0, 2, 3, 8, 5, 6, 7, 16, 9, 20, 11, 12, 13, 28, 15, 32, 17, 18, 19, 20, 21, 44, 23, 24, 25, 52, 27, 28, 29, 30, 31, 64, 33, 68, 35, 72, 37, 76, 39, 40, 41, 42, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 54, 55, 56, 57, 116, 59, 120, 61, 124, 63, 128, 65, 66

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

a(n) / n = 1 for numbers n from A242482, a(n) / n = 2 for numbers n from A242483.
If there are any odd multiply-perfect numbers n > 1 then a(n) = 0.
Possible values of a(n) in increasing order = A242485. Numbers m such that a(n) = m has no solution = A242486.

			a(8) = (8*(8+1)/2) mod 8 + sigma(8) mod 8 + antisigma(8) mod 8 = 36 mod 8 + 15 mod 8 + 21 mod 8 = 4 + 7 + 5 = 16.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A242481, A242482, A242483, A242484, A242485, A242486.

[((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n)): n in [1..1000]]

a(n) = A142150(n) + A054024(n) + A229110(n) = (A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n).

a(n) = A242481(n) * n.

A242481 a(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

a(1) = 0. If there is no odd multiply-perfect number then a(n) = 1 or 2 for n >= 2. See A242482 = numbers m such that a(n) = 1, A242483 = numbers m such that a(n) = 2. If there are any odd multiply-perfect numbers m > 1 then a(m) = 0.

			a(8) = [(8*(8+1)/2) mod 8 + sigma(8) mod 8 + antisigma(8) mod 8] / 8 = (36 mod 8 + 15 mod 8 + 21 mod 8) / 8 = (4 + 7 + 5 ) / 8 = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A242480, A242482, A242483, A242484, A242485, A242486.

[((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n))div n: n in [1..1000]]

a(n) = (A142150(n) + A054024(n) + A229110(n)) / n = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) / n.

a(n) = A242480(n) / n.

A242482 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 42, 43, 45, 47, 49, 51, 53, 54, 55, 56, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 97, 99

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

Numbers n such that A242480(n) = (1/2*n*(n+1)) mod n + sigma(n) mod n + antisigma(n) mod n = (A142150(n) + A054024(n) + A229110(n)) = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) = n. Numbers n such that A242481(n) = (A142150(n) + A054024(n) + A229110(n)) / n = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) / n = 1.
Conjecture: with number 1 complement of A242483.
Supersequence of primes (A000040).
If there is no odd multiply-perfect number, then:
(1) a(n) = union of odd numbers >= 3 and even numbers from A239719.
(2) a(n) = supersequence of odd numbers (A005408).

			6 is in sequence because [(6*(6+1)/2) mod 6 + sigma(6) mod 6 + antisigma(6) mod 6] / 6 = (21 mod 6 + 12 mod 6 + 9 mod 6) / 6 = (3 + 0 + 3 ) / 6 = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..5000

Cf. A242480, A242481, A242483, A242484, A242485, A242486.

[n: n in [1..1000] | n eq ((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n))]

A242483 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 2.

Original entry on oeis.org

4, 8, 10, 14, 16, 22, 26, 32, 34, 36, 38, 44, 46, 48, 50, 52, 58, 60, 62, 64, 68, 72, 74, 76, 82, 84, 86, 90, 92, 94, 96, 98, 106, 108, 110, 116, 118, 122, 124, 128, 130, 132, 134, 136, 142, 144, 146, 148, 152, 154, 156, 158, 164, 166, 168, 170, 172, 178, 182

Offset: 1

Jaroslav Krizek, May 16 2014

nonn

Numbers n such that A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = (A142150(n) + A054024(n) + A229110(n)) = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) = 2n. Numbers n such that A242481(n) = (A142150(n) + A054024(n) + A229110(n)) / n = ((A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n)) / n = 2.

Conjecture: with number 1 complement of A242482.

			8 is in sequence because [(8*(8+1)/2) mod 8 + sigma(8) mod 8 + antisigma(8) mod 8] / 8 = (36 mod 8 + 15 mod 8 + 21 mod 8) / 8 = (4 + 7 + 5 ) / 8 = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..5000

Cf. A242480, A242481, A242482, A242484, A242485, A242486.

[n: n in [1..1000] | 2 eq (((n*(n+1)div 2 mod n + SumOfDivisors(n) mod n + (n*(n+1)div 2-SumOfDivisors(n)) mod n)))div n]

A205523 Numbers k such that gcd(k, sigma(k)) == sigma(k) (mod k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 12, 13, 17, 18, 19, 20, 23, 24, 28, 29, 31, 37, 40, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 83, 88, 89, 97, 101, 103, 104, 107, 109, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 180, 181, 191, 193, 196, 197, 199

Offset: 1

Jaroslav Krizek, Jan 28 2012

nonn

Numbers m such that A009194(m) = gcd(m, A000203(m)) = A000203(m) mod m = A054024(m).

Complement of A205524. Union of primes (A000040) and composite numbers from A205525.

			Number 24 is in sequence because gcd(24, sigma(24)) = (sigma(24)=60) mod 24 = 12.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A000040, A009194, A054024, A205524, A205525.

Select[Range[300], Mod[GCD[#, DivisorSigma[1, #]] - DivisorSigma[1, #], #] == 0 &]

isok(n) = (gcd(n, sigma(n)) % n) == (sigma(n) % n); \\ Michel Marcus, Dec 22 2017

Corrected by T. D. Noe, Feb 03 2012

A242485 Possible values of A242480(n) in increasing order.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 83

Offset: 1

Jaroslav Krizek, May 27 2014

nonn

A242480(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n = A142150(n) + A054024(n) + A229110(n) = (A000217(n) mod n) + (A000203(n) mod n) + (A024816(n) mod n).

Supersequence of odd numbers > 1. Complement of A242486.

			16 is in the sequence because there is a number m such that A242480(m) = 16; m = 8.

Cf. A242480, A242481, A242482, A242483, A242484, A242486.

cp's OEIS Frontend

A076496 Numbers k such that sigma(k) == 12 (mod k).

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A045769 Numbers k such that sigma(k) == 4 (mod k).

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A088834 Numbers k such that sigma(k) == 6 (mod k).

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A108775 a(n) = floor(sigma(n)/n).

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A242480 a(n) = (n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n.

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A242481 a(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n.

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A242482 Numbers n such that A242481(n) = ((n*(n+1)/2) mod n + sigma(n) mod n + antisigma(n) mod n) / n = 1.

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