A076496 -id:A076496 - cp's OEIS frontend

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

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

A054024 Sum of the divisors of n reduced modulo n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 4, 8, 1, 4, 1, 10, 9, 15, 1, 3, 1, 2, 11, 14, 1, 12, 6, 16, 13, 0, 1, 12, 1, 31, 15, 20, 13, 19, 1, 22, 17, 10, 1, 12, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 12, 17, 8, 23, 32, 1, 48, 1, 34, 41, 63, 19, 12, 1, 58, 27, 4, 1, 51, 1, 40, 49, 64, 19, 12, 1, 26, 40

Offset: 1

Asher Auel, Jan 19 2000

If a(n) = 0, then n is a multiply-perfect number (A007691). - Alonso del Arte, Mar 30 2014

			a(12) = 4 because sigma(12) = 28 and 28 == 4 (mod 12).
a(13) = 1 because 13 is prime.
a(14) = 10 because sigma(14) = 24 and 24 == 10 (mod 14).

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe, terms 1001..20000 from Alois P. Heinz).
Walter Nissen, Abundancy : Some Resources.

Cf. A000203 (sigma), A005114 (untouchable numbers), A007691 (positions of 0's), A045768, A045769, A088834, A045770, A076496, A159907.

a054024 n = mod (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

with(numtheory): seq(sigma(i) mod i, i=1..100);

Table[Mod[DivisorSigma[1, n], n], {n, 80}] (* Alonso del Arte, Mar 30 2014 *)

a(n)=sigma(n)%n \\ Charles R Greathouse IV, Nov 04 2014

a(n) = sigma(n) mod n.

a(p) = 1 for p prime.

A045770 Numbers k such that sigma(k) == 8 (mod k).

Original entry on oeis.org

1, 7, 10, 49, 56, 368, 836, 11096, 17816, 45356, 77744, 91388, 128768, 254012, 388076, 2087936, 2291936, 13174976, 29465852, 35021696, 45335936, 120888092, 260378492, 381236216, 775397948, 3381872252, 4856970752, 6800228816, 8589344768, 44257207676, 114141404156, 1461083549696, 1471763808896, 2199013818368

Offset: 1

Dan Hoey

nonn

Every number of the form 2^(j-1)*(2^j - 9), where 2^j - 9 is prime, is a term. - Jon E. Schoenfield, Jun 02 2019

If m is a term of A045768 with gcd(m,3) = 1 and sigma(m) = 3*q*m + 2 for some integer q, then 3*m is a term of this sequence since sigma(3*m) = 4*q*(3*m) + 8. Some other large terms: 36893488108764397568, 877615520070055755776, 1700388548189538291286016, 85954979333046510417991676, 2081228720695521934665574252544. - Max Alekseyev, May 25 2025

Max Alekseyev, Table of n, a(n) for n = 1..41 (first 36 terms from Jud McCranie)

Cf. A054024, A045768, A045769, A088834, A076496.

q:= k-> nops(map(x-> x mod k, {8, numtheory[sigma](k)}))=1:
select(q, [$1..100000])[];  # Alois P. Heinz, Apr 07 2025

Select[Range[1000000], Mod[DivisorSigma[1, #] - 8, #] == 0 &] (* Pontus von Brömssen, Apr 07 2025 *)

isok(k) = Mod(sigma(k),k) == 8; \\ Pontus von Brömssen, Apr 07 2025

a(18)-a(26) from T. D. Noe, Apr 06 2011
Initial term 1 added and a(27)-a(31) from Donovan Johnson, Mar 01 2012
a(32)-a(34) from Giovanni Resta, Apr 02 2014
Term a(2)=7 inserted by Pontus von Brömssen, Apr 07 2025

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

Original entry on oeis.org

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

A067702 Numbers k such that sigma(k) == 0 (mod k+2).

Original entry on oeis.org

12, 70, 88, 180, 1888, 4030, 5830, 32128, 521728, 1848964, 8378368, 34359083008, 66072609790, 549753192448, 259708613909470, 2251799645913088

Offset: 1

Benoit Cloitre, Feb 05 2002

If 2^i-5 is prime for i > 2 then let x = (2^i-5)*2^(i-1). Then sigma(x)=2*(x+2), so x is in the sequence. There are other terms that are not of this form. - Jud McCranie, Jan 12 2019

Contains terms of A088832, terms m of A088834 with (sigma(m)-6)/m = 3, terms m of A045770 with (sigma(m)-8)/m = 4, terms m of A076496 with (sigma(m)-12)/m = 6. - Max Alekseyev, May 26 2025

			sigma(180) = 546 = 3(180+2), so 180 is in the sequence.

Contains subsequence A088832.

Cf. A045768, A156560, A059608, A045770, A076496, A088834.

Select[Range[84*10^5],Divisible[DivisorSigma[1,#],#+2]&] (* Harvey P. Dale, May 11 2018 *)

isok(n) = sigma(n) % (n+2) == 0; \\ Michel Marcus, Nov 22 2013

a(9)-a(11) from Michel Marcus, Nov 22 2013
a(12)-a(13) from Jud McCranie, Jan 12 2019
a(14) from Jud McCranie, Jan 13 2019
a(15) from Jud McCranie, Dec 02 2019
a(16) from Max Alekseyev, May 26 2025

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

A141545 Numbers k whose abundance is 12: sigma(k) - 2*k = 12.

Original entry on oeis.org

24, 30, 42, 54, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 304, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

nonn

Numbers k such that sigma(k) = 2k + 12. - Wesley Ivan Hurt, Jul 11 2013
Any term x = a(m) can be combined with any term y = A141549(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2.  Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one.  So far, these two sequences have not produced an amicable pair.  However, if one is ever found, then it will exhibit y-x = 12. - Timothy L. Tiffin, Sep 13 2016
From Tomohiro Yamada, Jan 01 2023: (Start)
6p belongs to this sequence if p > 3 is prime since sigma(6p) = 12(p + 1) = 12p + 12.  Moreover, 2^m * (2^(m+1) - 13) is also a term of this sequence if 2^(m+1) - 13 is prime (m+1 is a term of A096818) since sigma(2^m * (2^(m+1) - 13)) = (2^(m+1) + 1) * (2^(m+1) - 13) = 2^(m+1) * (2^(m+1) - 13) + 12.  So 24, 304, 127744, 33501184, and 8589082624 also belong to this sequence.
Problem: is 54 the only term of this sequence which is of neither type given above? (End)

			30 is in the sequence since sigma(30) = sigma(2*3*5) = sigma(2)*sigma(3)*sigma(5) = 3*4*6 = 72 = 2(30)+12.  Since this is the second such number whose abundance is 12, a(2) = 30. - _Wesley Ivan Hurt_, Jul 11 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A000203, A005101, A141549 (deficiency 12).

Cf. A076496 (sigma(k) - a*k = 12).

[n: n in [1..1400] | (SumOfDivisors(n)-2*n) eq 12]; // Vincenzo Librandi, Sep 14 2016

lst={};Do[If[n==Plus@@Divisors[n]-n-12,AppendTo[lst,n]],{n,10^4}];Print[lst];
Select[Range[1, 10^4], DivisorSigma[1, #] - 2 # == 12 &] (* Vincenzo Librandi, Sep 14 2016 *)

is(n)=sigma(n)==2*n+12 \\ Charles R Greathouse IV, Feb 21 2017

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

A096821 Solutions to Mod[sigma(x),x]=12 of the form p*(p+13)/2 where p is a prime of form=2^j-13, of which j exponents are listed in A096818.

Original entry on oeis.org

24, 304, 127744, 33501184, 8589082624, 10384593717069654320312270165377024, 822752278660603021077484591278411581166520461101278617407586304, 15177100720513508366558296147058741458142670970377727126573378340391656803557965824

Offset: 1

Labos Elemer, Jul 13 2004

nonn

The relevant A084306 includes additional solution like 21, while A076496 contains solutions of 6k form too.

			8th term is (2^136)*(-13+2^137) with 83 decimal digits; n=4: a[4]=(2^3)*(16-13)=24;

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A084306, A096518, A000203, A076496.

A216697 Numbers n such that sigma(n) mod n = 12, n is divisible by 6, but n/6 is not prime.

Original entry on oeis.org

24, 54, 780, 2352, 430272, 184773312

Offset: 1

Michel Marcus, Sep 15 2012

nonn

Motivated by A076496 comment: if n=6p, p>3 prime, then Mod(sigma(n),n)=12. So this sequence is included in A076496, but not in A084306.

Next term > 10^11. - Donovan Johnson, Sep 27 2012

			sigma(24) = 60 = 2*24+12, but 24/6=4 is not prime.

Cf. A076496, A084306.

isOk(n) = { return ((n % 6 == 0) && (! isprime(n/6)) && (sigma(n) % n == 12));}

cp's OEIS Frontend

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

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A054024 Sum of the divisors of n reduced modulo n.

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A045770 Numbers k such that sigma(k) == 8 (mod k).

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

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A067702 Numbers k such that sigma(k) == 0 (mod k+2).

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

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A141545 Numbers k whose abundance is 12: sigma(k) - 2*k = 12.

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Magma