A002961 -id:A002961 - cp's OEIS frontend

A054003 tau(n+1) - tau(n) where n and n+1 have the same sum of divisors.

0, 2, -4, 0, 4, 0, 0, 8, 6, -4, -6, 6, -8, -8, 0, 4, 0, 4, 0, 8, 8, -8, 0, -4, 0, 8, -8, 0, -12, 12, -8, 8, 0, -8, -8, -4, -12, 24, -8, 0, 8, 0, -8, -4, -8, -8, 0, 16, -4, -4, 4, 0, 0, -28, 0, 0, -20, -4, 24, 0, -16, 8, 8, -8, -8, 12, -16, 0, -40, 40, -8, 8, 0, 0, 0, 40, -8, 0, 40

Offset: 1

Asher Auel, Jan 12 2000

sign

Amiram Eldar, Table of n, a(n) for n = 1..10135 (from the b-file at A002961)

Cf. A000005, A000203, A002961, A053249, A054002.

[#Divisors(n+1)-#Divisors(n):n in [1..5000000]| SumOfDivisors(n) eq SumOfDivisors(n+1)]; // Marius A. Burtea, Sep 07 2019

a(n) = A054002(n) - A053249(n).

More terms from Naohiro Nomoto, Jun 23 2001

A077086 Remainder when sigma(n+1) is divided by sigma(n).

Original entry on oeis.org

0, 1, 3, 6, 0, 8, 7, 13, 5, 12, 4, 14, 10, 0, 7, 18, 3, 20, 2, 32, 4, 24, 12, 31, 11, 40, 16, 30, 12, 32, 31, 48, 6, 48, 43, 38, 22, 56, 34, 42, 12, 44, 40, 78, 72, 48, 28, 57, 36, 72, 26, 54, 12, 72, 48, 80, 10, 60, 48, 62, 34, 8, 23, 84, 60, 68, 58, 96, 48, 72, 51, 74, 40, 10

Offset: 1

Labos Elemer, Oct 31 2002

nonn

			a(7) = 7 since sigma(7) = 8, sigma(8) = 15, and 15 = 1*8 + 7.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A000668 (fixed points), A002961, A067081.

a[n_]:=Mod[DivisorSigma[1,n+1],DivisorSigma[1,n]]; Array[a,74] (* Stefano Spezia, Jan 24 2025 *)

a(n) = sigma(n+1) % sigma(n); \\ Michel Marcus, Dec 26 2013

a(n) = Mod(A000203(n+1), A000203(n)).

A132585 Numbers k such that sigma(k)-k-1 divides sigma(k+1)-k-2, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

Original entry on oeis.org

25, 49, 799, 899, 32399, 292681, 1492995736325809

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2007

The banal case of ratio equal to zero is excluded. In fact if k+1 is a prime than sigma(k+1)-k-2=0. Therefore the ratio with sigma(k)-k-1 is equal to zero. Is this sequence finite?
a(7) <= 1492995736325809. [From Donovan Johnson, Aug 31 2008]
a(7) > 10^13. - Giovanni Resta, Jul 11 2013
No other terms < 2.7*10^15. - Jud McCranie, Jul 26 2025

			k=25 -> sigma(25)= 1+5+25 -> sigma(k)-k-1=5
k+1=26 -> sigma(26)= 1+2+13+26 -> sigma(k+1)-k-2=2+13=15
15/5 = 3 (integer > 0)

Cf. A000203, A002961, A058072, A058073, A132586.

with(numtheory); P:=proc(n) local a,i; for i from 1 by 1 to n do if sigma(i)-i-1>0 then a:=(sigma(i+1)-i-2)/(sigma(i)-i-1); if a>0 and trunc(a)=a then print(i); fi; fi; od; end: P(100000);

a(6) from Donovan Johnson, Aug 31 2008

a(7) by Jud McCranie, Jul 26 2025

A132586 Numbers k such that sigma(k+1)-k-2 divides sigma(k)-k-1, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

Original entry on oeis.org

8, 24, 8925, 32445, 118540859325

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2007

The banal case of ratio equal to zero is excluded. In fact if k is a prime than sigma(k)-k-1=0. Therefore the ratio with sigma(k+1)-k-2 is equal to zero. Is this sequence finite?
a(6), if it exists, is larger than 10^13. - Giovanni Resta, Jul 13 2015
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			n=8 -> sigma(8)=1+2+4+8 -> sigma(n)-n-1=2+4=6.
n+1=9 -> sigma(9)=1+3+9 -> sigma(n+1)-n-2=3.
6/3 = 2 (integer >0)

Cf. A002961, A058072, A058073, A132585.

with(numtheory); P:=proc(n) local a,i; for i from 1 by 1 to n do if sigma(i+1)-i-2>0 then a:=(sigma(i)-i-1)/(sigma(i+1)-i-2); if a>0 and trunc(a)=a then print(i); fi; fi; od; end: P(100000);

a(5) from Donovan Johnson, Aug 31 2008

A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).

Original entry on oeis.org

24, 215, 280, 1079, 947519, 1362239, 2230271, 14939999, 19720007, 32509439, 45581759, 45841247, 49436927, 78436511, 82842911, 101014631, 166828031, 225622151, 225757799, 250999559, 377129087, 554998751, 619606439, 846765431, 1204092287, 1302170687, 1710035711

Offset: 1

Michel Lagneau, Apr 04 2010

nonn

The equation sigma_2(m) = sigma_2(m + k) has infinitely many solutions where k >= 2 and k is even (J.-M. De Koninck).
From Amiram Eldar, Apr 19 2024: (Start)
De Koninck's proof is based on the assumption of Schinzel's hypothesis H. If q, r = q + 2, s = q^2 + q + 1, and p = q^2 + 3*q + 3 are all primes, then p*q is a term (the values of q+1 are the terms of A268043).
The equation sigma_2(m) = sigma_2(m + 1) has only one solution: m = 6. (End)

			For m=24, sigma_2(24) = sigma_2(26) = 850.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 118, entry 1079.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B13, pp. 103-104.

Amiram Eldar, Table of n, a(n) for n = 1..156
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Jean-Marie De Koninck, On the solutions of sigma_2(n) = sigma_2(n + l), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
Wikipedia, Schinzel's hypothesis H.

Cf. A001157, A002961, A007373, A175199, A268043.

with(numtheory):for n from 1 to 500000000 do:liste:= divisors(n) : s2 :=sum(liste[i]^2, i=1..nops(liste)):liste:=divisors(n+2):s3:=sum(liste[i]^2, i=1..nops(liste)):if s2 = s3 then print(n):else fi:od:

Select[Range[10^9], DivisorSigma[2,#] == DivisorSigma[2,#+2]&]

is(n) = sigma(n, 2) == sigma(n + 2, 2); \\ Amiram Eldar, Apr 19 2024

lista(mmax) = {my(s1 = sigma(1, 2), s2 = sigma(2, 2), s3, s4); forstep(m = 3, mmax, 2, s3 = sigma(m, 2); s4 = sigma(m+1, 2); if(s1 == s3, print1(m - 2, ", ")); if(s2 == s4, print1(m - 1, ", ")); s1 = s3; s2 = s4);} \\ Amiram Eldar, Apr 19 2024

a(25)-a(27) from Donovan Johnson, Apr 14 2013

A175875 Numbers k such that sigma(k+3) = 3*sigma(k).

Original entry on oeis.org

3, 30237, 292317, 3116957, 4044037, 6902277, 73442597, 115767957, 137410557, 416776357, 526908197, 541579497, 695350653, 758403357, 1105731897, 1178082357, 1233277917, 1309742557, 1866261597, 1998267037, 2161411557, 2563416237, 2750761437, 2873771997, 2892203997, 3331848517, 3621735037, 3758847117

Offset: 1

Zak Seidov, Oct 06 2010

nonn

a(1) = A175874(3).

Giovanni Resta, Table of n, a(n) for n = 1..180 (terms < 10^12)

Cf. A000203, A002961, A175874, A272027 (3*sigma(n)).

[n: n in [1..10^8] | SumOfDivisors(n+3) eq 3*SumOfDivisors(n)]; // Vincenzo Librandi, Jul 28 2017

isok(n) = sigma(n+3) == 3*sigma(n); \\ Michel Marcus, Oct 19 2013

a(8)-a(21) from Donovan Johnson, Oct 11 2010
a(22)-a(24) from Donovan Johnson, Sep 10 2012
a(25) from Zak Seidov, Jul 07 2013
a(26)-a(28) from Chai Wah Wu, Jul 27 2017

A192282 Numbers n such that n and n+1 have same sum of anti-divisors.

Original entry on oeis.org

1, 8, 17, 120, 717, 729, 957, 8097, 10785, 12057, 35817, 44817, 52863, 58677, 59757, 76759, 95397, 102957, 114117, 119337, 182157, 206097, 215997, 230037, 253977, 263877, 269277, 271797, 295377, 321417, 402657, 435477, 483117, 485637, 510837, 586797, 589317

Offset: 1

Paolo P. Lava, Jul 27 2011

nonn

Like A002961 but using anti-divisors.

Curiously 957 and 958 have same sum of divisors and same sum of anti-divisors.

			Anti-divisors of 717 are 2, 5, 6, 7, 35, 41, 205, 287, 478 and their sum is 1066.
Anti-divisors of 718 are  3, 4, 5, 7, 35, 41, 205, 287, 479 and their sum is 1066.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A002961, A066272, A192283.

with(numtheory);
P:=proc(n)
local a,b,i,k;
b:=2;
for i from 4 to n do
  a:=0;
  for k from 2 to i-1 do
    if abs((i mod k)- k/2) < 1 then a:=a+k; fi;
  od;
  if a=b then print(i-1); fi;
  b:=a;
od;
end:
P(200000);

Initial term a(1)=1 inserted, a(2)=9 through a(20)=119337 verified, and a(21)-a(28) added by John W. Layman, Aug 04 2011

A217791 Numbers k such that sigma(k) = 3*sigma(k+1).

Original entry on oeis.org

180, 12000, 30996, 47940, 66780, 102816, 128040, 234300, 494088, 712272, 1133088, 1408212, 1623072, 1692768, 1896336, 1925196, 2024760, 2388720, 2529090, 2836008, 3423120, 3724320, 3822360, 4628760, 4750920, 7219608, 7359912, 7603488, 7749060

Offset: 1

Paolo P. Lava, Mar 25 2013

nonn

			47940 is in the sequence because sigma(47940)=145152, sigma(47941)=48384, and 145152=3*48384.
7749060 is in the sequence because sigma(7749060)=24192000, sigma(7749061)=8064000, and 24192000=3*8064000.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000203, A002961, A058073, A067081, A077087, A163193, A272027 (3*sigma(n)).

[n: n in [1..10^7] | SumOfDivisors(n) eq 3*SumOfDivisors(n+1)]; // Bruno Berselli, Mar 25 2013

A217791:=proc(q) local n;
for n from 1 to q do if sigma(n)=3*sigma(n+1) then print(n); fi; od; end:
A217791(10^10);

Position[Partition[DivisorSigma[1,Range[78*10^5]],2,1],?(#[[1]] == 3#[[2]]&), {1},Heads->False]//Flatten (* _Harvey P. Dale, Oct 17 2016 *)

More terms from Bruno Berselli, Mar 25 2013

A223136 Numbers n such that sigma(n+1) - sigma(n) = k*n for some integer k, where sigma(n) = A000203 (sum of divisors of n).

Original entry on oeis.org

1, 3, 7, 14, 31, 127, 206, 532, 954, 957, 1334, 1364, 1634, 2685, 2974, 4364, 8191, 14841, 18873, 19358, 20145, 24957, 33998, 36566, 42818, 56564, 64665, 74918, 79826, 79833, 84134, 92685, 104944, 109214, 111506, 116937, 122073, 131071, 138237, 147454, 161001

Offset: 1

Jaroslav Krizek, May 01 2013

nonn

Supersequence of A000668 for k=1 (Mersenne primes), A067803 for k=-1 (numbers n such that sigma(n) - sigma(n+1) = n) and A002961 for k=0 (numbers n such that n and n+1 have same sum of divisors). For number 1 is k=2.

Corresponding values of integers k: 2, 1, 1, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,....

			Number 532 is in sequence because sigma(533) - sigma(532) = 588 - 1120 = -532 = (-1) * 532; k = -1.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000668, A067803, A002961, A000203.

Select[Range[10000], IntegerQ[(DivisorSigma[1, # + 1] - DivisorSigma[1, #])/#] &] (* T. D. Noe, May 02 2013 *)

Extended by T. D. Noe, May 02 2013

A225757 Table of consecutive numbers with the same sum of divisors.

Original entry on oeis.org

14, 15, 206, 207, 957, 958, 1334, 1335, 1364, 1365, 1634, 1635, 2685, 2686, 2974, 2975, 4364, 4365, 14841, 14842, 18873, 18874, 19358, 19359, 20145, 20146, 24957, 24958, 33998, 33999, 36566, 36567, 42818, 42819, 56564, 56565, 64665, 64666, 74918, 74919, 79826

Offset: 1

Jean-François Alcover, May 15 2013

Are 3 consecutive terms possible? There are none less than 10^12. See A002961. - T. D. Noe, May 15 2013

			Sequence begins:
14, 15;
206, 207;
957, 958;
1334, 1335;
etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000 (first 226 terms from T. D. Noe)

Cf. A225756 (same number of divisors), A225758 (same number and sum of divisors), A002961 (first number of each pair).

sel = Select[Range[100000], DivisorSigma[1, #] == DivisorSigma[1, # + 1] &]; Union[sel, sel + 1]
Flatten[SequencePosition[DivisorSigma[1,Range[80000]],{x_,x_}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 13 2017 *)

v=List();t=[1,3];for(n=3,1e6,t=[t[2],sigma(n)];if(t[1]==t[2],listput(v,n-1);listput(v,n)));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, May 15 2013

cp's OEIS Frontend

A054003 tau(n+1) - tau(n) where n and n+1 have the same sum of divisors.

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A077086 Remainder when sigma(n+1) is divided by sigma(n).

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A132585 Numbers k such that sigma(k)-k-1 divides sigma(k+1)-k-2, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

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A132586 Numbers k such that sigma(k+1)-k-2 divides sigma(k)-k-1, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

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A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).

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A175875 Numbers k such that sigma(k+3) = 3*sigma(k).

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A192282 Numbers n such that n and n+1 have same sum of anti-divisors.

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A217791 Numbers k such that sigma(k) = 3*sigma(k+1).