A045768 -id:A045768 - cp's OEIS frontend

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

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

A055708 Numbers n such that n - 1 | sigma(n).

Original entry on oeis.org

2, 3, 10, 136, 32896, 2147516416

Offset: 1

Robert G. Wilson v, Dec 04 2000

a(7) > 10^13. - Giovanni Resta, Jul 13 2015

Cf. A000203, A045768, A056006.

Do[ If[ Mod[ DivisorSigma[1, n], n - 1] == 0, Print[n] ], {n, 2, 10^8} ]

is(n)=sigma(n)%(n-1)==0 \\ Anders Hellström, Aug 15 2015

a(6) from Donovan Johnson, Nov 15 2009

A056006 Numbers k such that k | sigma(k) + 2.

Original entry on oeis.org

1, 3, 10, 136, 32896, 2147516416

Offset: 1

Robert G. Wilson v, Jul 24 2000

n | sigma(n) gives the multi-perfect numbers A007691, n | sigma(n)+1 if n is a power of 2 (A000079).
This contains A191363 as subsequence, so for any Fermat prime F(k) = 2^2^k+1, the triangular number A000217(2^2^k)=(F(k)-1)*F(k)/2 is in this sequence. See also A055708 which is identical up to the first term. - M. F. Hasler, Oct 02 2014
a(7) > 10^13. - Giovanni Resta, Jul 13 2015
a(7) > 10^18. - Max Alekseyev, May 27 2025

Cf. A000203, A045768, A055708.

Do[If[Mod[DivisorSigma[1, n]+2, n]==0, Print[n]], {n, 1, 7*10^8}]

for(n=1,5e9,if((sigma(n)+2)%n==0,print1(n", "))) \\ Charles R Greathouse IV, Jun 01 2011

a(6) from Charles R Greathouse IV, Jun 01 2011

Edited by M. F. Hasler, Oct 02 2014

A117346 Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 10, 11, 13, 16, 17, 19, 20, 23, 28, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 70, 71, 73, 79, 83, 88, 89, 97, 101, 103, 104, 107, 109, 110, 113, 120, 127, 128, 131, 136, 137, 139, 149, 151, 152, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Sigma is the sum_of_divisors function.

			70 is in the sequence because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiperfect Number.

Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117347 through A117350.

asmlQ[n_]:=Module[{p=Mod[DivisorSigma[1,n],n]},If[p>n/2,p=n-p];p<=Log[n]];
Select[Range[200],asmlQ] (* Harvey P. Dale, Dec 25 2013 *)

First term prepended by Harvey P. Dale, Dec 25 2013

A117349 Near-multiperfects with primes, powers of 2 and 6 * prime excluded, abs(sigma(n) mod n) <= log(n).

Original entry on oeis.org

6, 10, 20, 28, 70, 88, 104, 110, 120, 136, 152, 464, 496, 592, 650, 672, 884, 1155, 1888, 1952, 2144, 4030, 5830, 8128, 8384, 8925, 11096, 17816, 18632, 18904, 30240, 32128, 32445, 32760, 32896, 33664, 45356, 70564, 77744, 85936, 91388, 100804, 116624

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma(n) really "near" a multiple of n, for n=9? Or n=18? Log is the natural logarithm. Sigma is the sum_of_divisors function.

			70 is a term because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.

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

Donovan Johnson, Table of n, a(n) for n = 1..180 (terms <= 10^12)
Eric Weisstein's World of Mathematics, Multiperfect Number

Cf. A045768, A045769, A045770, A077374, A087167, A087485.
Cf. A088007, A088008, A088009, A088010, A088011, A088012.
Cf. A117346, A117347, A117348, A117350.

sigma(n) = k*n + r, abs(r) <= log(n).

Offset corrected by Donovan Johnson, Oct 01 2012

A117350 Near-multiperfects with primes, powers of 2, 6 * prime and 2^n * prime excluded, abs(sigma(n) mod n) <= log(n).

Original entry on oeis.org

70, 110, 120, 650, 672, 884, 1155, 4030, 5830, 8925, 11096, 17816, 18632, 18904, 30240, 32445, 32760, 45356, 70564, 77744, 85936, 91388, 100804, 116624, 244036, 254012, 388076, 391612, 430272, 442365, 523776, 1090912, 1848964, 2178540

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near." E.g., is sigma (n) really "near" a multiple of n, for n=9? Or n=18? Sigma is the sum_of_divisors function.

			70 is in the sequence because sigma(70) = 144 = 2*70 + 4, while 4 < log(70) ~= 4.248.
The 2-perfect numbers are excluded because they are 2^n * prime.

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

Donovan Johnson, Table of n, a(n) for n = 1..114 (terms <= 10^12)
Eric Weisstein's World of Mathematics, Multiperfect Number.

Cf. A045768 through A045770, A077374, A087167, A087485, A088007 through A088012, A117346 through A117349.

Offset corrected by Donovan Johnson, Oct 01 2012

A076495 Smallest x such that sigma(x) mod x = n, or 0 if no such x exists.

Original entry on oeis.org

2, 20, 4, 9, 0, 25, 8, 10, 15, 14, 21, 24, 27, 22, 16, 26, 39, 208, 36, 34, 51, 38, 57, 112, 95, 46, 69, 48, 115, 841, 32, 58, 45, 62, 93, 660, 155, 1369, 162, 44, 63, 1681, 50, 82, 123, 52, 129, 60, 75, 94, 72, 352, 235, 90, 329, 84, 99, 68, 265, 96, 371, 118, 64, 76

Offset: 1

Labos Elemer, Oct 21 2002

nonn

At present, the 0 entry for n=5 is only a conjecture.
For n <= 1000, a(5) and a(898) are the only terms not found using x <= 10^11. - Donovan Johnson, Sep 20 2012
10^11 < a(898) <= 140729946996736. - Donovan Johnson, Sep 28 2013
a(898) > 10^13 and the same bound holds for a(5), if it exists. - Giovanni Resta, Apr 02 2014
a(5) > 1.5*10^14, if it exists. - Jud McCranie, Jun 02 2019

			n=1: a(1) = smallest prime = 2.
n=3: a(3) = 4 since sigma(4) mod 4 = 7 mod 4 = 3.
n=5: Very difficult case (see Comments section).

Donovan Johnson, Table of n, a(n) for n = 1..1000
Carl Pomerance, On the congruences σ(n) ≡ a (mod n) and n ≡ a (mod φ(n)), Acta Arithmetica 26:3 (1974-1975), pp. 265-272. (See theorem 4.)

Cf. A045768, A045769, A045770, A054024.

f[x_] := s=Mod[DivisorSigma[1, n], n]; t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000}]; t

a(n)=my(k);while(sigma(k++)%k!=n,);k \\ Charles R Greathouse IV, Dec 28 2013

A117347 Near-multiperfects with primes excluded, abs(sigma(m) mod m) <= log(m).

Original entry on oeis.org

4, 6, 8, 10, 16, 20, 28, 32, 64, 70, 88, 104, 110, 120, 128, 136, 152, 256, 464, 496, 512, 592, 650, 672, 884, 1024, 1155, 1888, 1952, 2048, 2144, 4030, 4096, 5830, 8128, 8192, 8384, 8925, 11096, 16384, 17816, 18632, 18904, 30240, 32128, 32445, 32760, 32768

Offset: 1

Walter Nissen, Mar 09 2006

nonn

Sequences A117346 through A117350 are an attempt to improve on sequences A045768 through A045770, A077374, A087167, A087485 and A088007 through A088012 and related sequences (but not to replace them) by using a more significant definition of "near". E.g., is sigma(n) (where sigma is the sum-of-divisors function) really "near" a multiple of n, for n = 9? Or n = 18?

			70 is a term because sigma(70) = 144 = 2 * 70 + 4, while 4 < log(70) ~= 4.248.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Nissen, Near Multiperfects.
Eric Weisstein's World of Mathematics, Multiperfect Number

Cf. A045768, A045769, A045770, A077374, A087167, A087485.
Cf. A088007, A088008, A088009, A088010, A088011, A088012.
Cf. A117346, A117348, A117349, A117350.

sigma(m) = k * m + r, abs(r) <= log(m).

Offset corrected by Amiram Eldar, Mar 05 2020

A181597 (N\{4})-perfect numbers, i.e., numbers m for which sigma(m)-4 = 2m, if 4|m, otherwise sigma(m) = 2m.

Original entry on oeis.org

6, 12, 88, 1888, 32128, 521728, 1848964, 8378368, 34359083008, 549753192448, 2251799645913088, 9223372026117357568, 2361183241263023915008

Offset: 1

Vladimir Shevelev, Nov 01 2010, Nov 03 2010

Or union of {6}, near-perfect numbers m (cf. A181595) for which d(m)=4, and all odd perfect numbers (if they exist). Note that (N\{2})-perfect numbers are numbers for which sigma(m)-2=2m, if m is even, and sigma(m)=2m, if m is odd. They are all even numbers of A045768 and all odd perfect numbers (if they exist).

Besides odd perfect numbers (if they exist), this sequence is formed by 6 followed by the multiples of 4 from A088832. - Max Alekseyev, Sep 02 2025

			88 is in the sequence since sigma(88) = 180 and 180 - 4 = 2*88.

Cf. A000396, A005101, A005820, A088832, A118372, A153501, A181595, A181596.

Invalid term removed and a(8)-a(10) from Donovan Johnson, Sep 14 2013

a(11)-a(13) from Max Alekseyev, Sep 02 2025

A084306 Numbers x such that sigma(x) mod x = 12 and x is not divisible by 6. Singular solutions mentioned in A076496.

Original entry on oeis.org

121, 304, 127744, 33501184, 8589082624

Offset: 1

Labos Elemer, Jun 11 2003

nonn

If n = P*q, where P is a multiple perfect number and q is prime so that gcd(P,q) = 1, then sigma(n) = kn(q+1). Consequently sigma(n) = knq + kn sigma(n) mod n = kn. Such values of n are regular solutions to this and analogous cases. Here, not these but the additional eccentric solutions are collected. Cf. A076496.
a(6) > 10^11. - Donovan Johnson, Sep 20 2012
If p = 2^k - 13 > 3 is a prime number, then 2^(k-1)*p is a term. This happens for k = 5, 9, 13, 17, 57, 105, 137, 3217, ... (A096818). - Giovanni Resta, Apr 01 2014

			n = 33501184 = 4096*8179; sigma(n) = 2n + 12 = 67002380.

Cf. A076496, A076495, A000396, A045768-A045770, A007691, A096818.

Do[s=Mod[DivisorSigma[1, n], n]; If[IntegerQ[n/100000], Print[{n}]]; If[Equal[s, 12]&&!Equal[Mod[n, 6], 0], Print[n]], {n, 1, 100000000}]

a(5) from Donovan Johnson, Sep 20 2012

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A055708 Numbers n such that n - 1 | sigma(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A056006 Numbers k such that k | sigma(k) + 2.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A117346 Near-multiperfects: numbers m such that abs(sigma(m) mod m) <= log(m).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A117349 Near-multiperfects with primes, powers of 2 and 6 * prime excluded, abs(sigma(n) mod n) <= log(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A117350 Near-multiperfects with primes, powers of 2, 6 * prime and 2^n * prime excluded, abs(sigma(n) mod n) <= log(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A076495 Smallest x such that sigma(x) mod x = n, or 0 if no such x exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A117347 Near-multiperfects with primes excluded, abs(sigma(m) mod m) <= log(m).

Views

Author