A015867 -id:A015867 - cp's OEIS frontend

A002961 Numbers k such that k and k+1 have same sum of divisors.

14, 206, 957, 1334, 1364, 1634, 2685, 2974, 4364, 14841, 18873, 19358, 20145, 24957, 33998, 36566, 42818, 56564, 64665, 74918, 79826, 79833, 84134, 92685, 109214, 111506, 116937, 122073, 138237, 147454, 161001, 162602, 166934

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

For the values of n < 2*10^10 in this sequence, sigma(n)/n is between 1.5 and 2.25. - T. D. Noe, Sep 17 2007
Whether this sequence is infinite is an unsolved problem, as noted in many of the references and links. - Franklin T. Adams-Watters, Jan 25 2010
144806446575 is the first term for which sigma(n)/n > 2.25. All n < 10^12 have sigma(n)/n > 3/2. - T. D. Noe, Feb 18 2010
A053222(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2011
Numbers n such that n + 1 = antisigma(n+1) - antisigma(n), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n. Example for n = 14: 15 = antisigma(15) - antisigma(14) = 96 - 81. - Jaroslav Krizek, Nov 10 2013
Up to 10^13, the value of the sigma(n)/n varies between 1417728000/945151999 (attained for n = 2835455997) and 2913242112/1263730145 (for n = 5174974943775). - Giovanni Resta, Feb 26 2014
Also numbers n such that A242962(n) = A242962(n+1), with A242962(n) = T(n) mod antisigma(n), where T(n) = A000217(n) is the n-th triangular number and antisigma(n) = A024816(n) is the sum of numbers less than n which do not divide n. - Jaroslav Krizek, May 29 2014
Guy and Shanks construct 5559060136088313 as a term of this sequence. - Michel Marcus, Dec 29 2014
Note that in all cases, n and n+1 are composite. - Zak Seidov, May 03 2016

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
R. K. Guy, Unsolved Problems in Theory of Numbers, Sect. B13.
W. Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..10135 (terms < 10^13; first 4804 terms from T. D. Noe)
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].
Jonathan Bayless and Paul Kinlaw, On repeated values of sigma and multiperfect numbers, Journal of Combinatorics and Number Theory, Vol. 7, No. 3 (2015), pp. 177-189.
Lourdes Benito, Solutions of the problem of Erdős-Sierpiński: sigma(n)=sigma(n+1), arXiv:0707.2190 [math.NT], 2007.
Richard Guy and Daniel Shanks, A Constructed Solution of sigma(n) = sigma(n+1), The Fibonacci Quarterly, Volume 12, Number 3, October 1974, 299.
A. Makowski, On Some Equations Involving Functions phi(n) and sigma(n), The American Mathematical Monthly, Vol. 67, No. 7 (Aug. - Sep., 1960), pp. 668-670.
N. J. A. Sloane & D. Singmaster, Correspondence 1972.
Andreas Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

Cf. A000203 (sigma function), A053215, A053249, A054004
Cf. A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647. - Reinhard Zumkeller, Nov 03 2010
Cf. A238380.

import Data.List (elemIndices)
a002961 n = a002961_list !! (n-1)
a002961_list = map (+ 1) $ elemIndices 0 a053222_list
-- Reinhard Zumkeller, Dec 28 2011

Flatten[Position[Partition[DivisorSigma[1,Range[170000]],2,1],{x_,x_}]] (* Harvey P. Dale, Aug 08 2011 *)
SequencePosition[DivisorSigma[1,Range[200000]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 06 2018 *)

t1=sigma(1);for(n=2,1e6,t2=sigma(n);if(t2==t1,print1(n-1", "));t1=t2) \\ Charles R Greathouse IV, Jul 15 2011

Sum_{n>=1} 1/a(n) is in the interval (0.080958, 610837) (Bayless and Kinlaw, 2015). - Amiram Eldar, Oct 15 2020

More terms from Jud McCranie, Oct 15 1997

A007373 Numbers k such that sigma(k+2) = sigma(k).

Original entry on oeis.org

33, 54, 284, 366, 834, 848, 918, 1240, 1504, 2910, 2913, 3304, 4148, 4187, 6110, 6902, 7169, 7912, 9359, 10250, 10540, 12565, 15085, 17272, 17814, 19004, 19688, 21410, 21461, 24881, 25019, 26609, 28124, 30592, 30788, 31484, 38210, 38982, 39786, 40310, 45354

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

Numbers k such that antisigma(k+2) - antisigma(k) = 2*k + 3, where antisigma(m) = A024816(m) = sum of nondivisors of m. - Jaroslav Krizek, Mar 17 2013

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 33, pp 12, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)
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].
A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Essentially the same as A055574.

Cf. A002961, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647. [From Reinhard Zumkeller, Nov 03 2010]

Flatten[Position[Partition[DivisorSigma[1,Range[300000]],3,1], {x_, , x}]] (* Harvey P. Dale, Aug 08 2011 *)
SequencePosition[DivisorSigma[1,Range[300000]],{x_,,x}][[All,1]] (* Harvey P. Dale, Nov 17 2022 *)

je=[]; for(n=1,10^5,a=sigma(n); b=sigma(n+2); if(a==b,je=concat(je,n))); je

More terms from Jason Earls, Jul 20 2001

A015861 Numbers k such that sigma(k) = sigma(k+3).

Original entry on oeis.org

382, 8922, 11935, 31815, 32442, 61982, 123795, 145915, 186615, 271215, 442362, 554715, 560382, 580635, 964535, 1191575, 1243375, 1369302, 1539942, 1642795, 2616702, 3141215, 3299062, 3556035, 3716895, 4201015, 5148294

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..300

Cf. A002961, A007373, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647.

Flatten[Position[Partition[DivisorSigma[1,Range[5200000]],4,1],{x_, y_, z_, x_}]] (* Harvey P. Dale, Aug 08 2011 *)
Select[Range[6000000], DivisorSigma[1, #]==DivisorSigma[1, # + 3] &] (* Vincenzo Librandi, Mar 10 2014 *)
SequencePosition[DivisorSigma[1,Range[5150000]],{x_,,,x_}][[;;,1]] (* Harvey P. Dale, Dec 01 2024 *)

is(n)=sigma(n)==sigma(n+3) \\ Charles R Greathouse IV, Mar 09 2014

A015865 Numbers k such that sigma(k) = sigma(k+5).

Original entry on oeis.org

6, 46, 1030, 2673, 4738, 4785, 10437, 14025, 20038, 20326, 23914, 28702, 31101, 39273, 39669, 41349, 41554, 44709, 46366, 55918, 68638, 74205, 93682, 94365, 96790, 103678, 115245, 115642, 124785, 169990, 182830, 185073, 207118, 214090

Offset: 1

Robert G. Wilson v

nonn

Using the method proposed by Guy and Shanks to construct solutions of sigma(k) = sigma(k + 1), it is possible to search for large terms of this sequence: If q = 3^(m+1) + 8 and p = (3^m*q - 5)/2 are primes, then 2*p is a term. This occurs for m = 0, 1 and 4, giving the terms 6, 46 and 20326. If q = 3^(m+1) - 22 and p = (3^m*q + 5)/2 are primes, then 3^m*q is a term. This occurs for m = 45 giving the term 26183890704263137277609197558886063678754201. In both cases there are no other solutions for m <= 10^4. - Amiram Eldar, May 29 2020

Donovan Johnson, Table of n, a(n) for n = 1..1000
Richard Guy and Daniel Shanks, A Constructed Solution of sigma(n) = sigma(n+1), The Fibonacci Quarterly, Vol. 12, No. 3 (1974), p. 299.

Cf. A000203, A002961, A007373, A015861, A015863, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647.

Select[Range[215000], DivisorSigma[1, #]==DivisorSigma[1, # + 5] &] (* Vincenzo Librandi, Mar 10 2014 *)
Position[Partition[DivisorSigma[1,Range[215000]],6,1],?(#[[1]] == #[[6]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 18 2021 *)

is(n)=sigma(n)==sigma(n+5) \\ Charles R Greathouse IV, Mar 09 2014

Corrected and extended by T. D. Noe, Oct 31 2006

A015863 Numbers k such that sigma(k) = sigma(k+4).

Original entry on oeis.org

51, 66, 115, 220, 319, 1003, 2585, 4024, 4183, 4195, 5720, 5826, 5959, 8004, 8374, 11659, 12367, 12561, 13581, 14338, 15365, 16116, 17840, 18718, 20541, 25130, 29393, 30170, 32665, 36516, 39913, 40660, 42423, 42922, 47841, 49762

Offset: 1

Robert G. Wilson v

nonn

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.

Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 130 terms from Paolo P. Lava)
A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

Cf. A002961, A007373, A015861, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647.

Select[Range[60000], DivisorSigma[1, #]==DivisorSigma[1, # + 4] &] (* Vincenzo Librandi, Mar 10 2014 *)
SequencePosition[DivisorSigma[1,Range[50000]],{x_,,,_,x_}][[;;,1]] (* Harvey P. Dale, Feb 13 2025 *)

is(n)=sigma(n)==sigma(n+4) \\ Charles R Greathouse IV, Mar 09 2014

A015866 Numbers k such that sigma(k) = sigma(k+6).

Original entry on oeis.org

20, 155, 182, 184, 203, 264, 621, 650, 702, 852, 893, 944, 1343, 1357, 2024, 2544, 2990, 4130, 4183, 4450, 5428, 5835, 6149, 6313, 6572, 8177, 8695, 11208, 11333, 11991, 12444, 12561, 12859, 13595, 14857, 16815, 18203, 18330, 18430, 19089

Offset: 1

Robert G. Wilson v

nonn

Up to 10^13, sigma(k-6) = sigma(k) = sigma(k+6) only for k = 33227, 604453 and 4223105512993. - Giovanni Resta, Mar 03 2014

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 20, pp 7, Ellipses, Paris 2008.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Donovan Johnson)
A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

Cf. A002961, A007373, A015861, A015863, A015865, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647.

Select[Range[20000], DivisorSigma[1, #]==DivisorSigma[1, # + 6] &] (* Vincenzo Librandi, Mar 10 2014 *)

is(n)=sigma(n)==sigma(n+6) \\ Charles R Greathouse IV, Mar 09 2014

A015876 Numbers k such that sigma(k) = sigma(k+8).

Original entry on oeis.org

15, 69, 87, 102, 132, 175, 230, 638, 689, 1127, 1349, 1392, 2006, 5170, 6680, 8366, 8390, 11652, 11918, 12128, 16748, 19511, 19829, 23318, 24597, 24734, 25122, 27162, 28676, 30730, 32864, 37391, 37436, 37901, 41082, 45925, 47487

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000
A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

Cf. A002961, A007373, A015861, A015863, A015865, A015866, A015867, A015877, A015880, A015881, A015882, A015883, A181647.

Select[Range[50000], DivisorSigma[1, #]==DivisorSigma[1, # + 8] &] (* Vincenzo Librandi, Mar 10 2014 *)

is(n)=sigma(n)==sigma(n+8) \\ Charles R Greathouse IV, Mar 09 2014

A015877 Numbers k such that sigma(k) = sigma(k+9).

Original entry on oeis.org

14, 16, 46, 446, 1146, 26766, 35805, 143605, 179086, 185946, 437745, 1187725, 1194646, 1327086, 1746946, 2201806, 2893605, 3003385, 3574725, 3730125, 4053586, 4928385, 5715325, 6220305, 7507946, 9423645, 9897186

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A002961, A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015880, A015881, A015882, A015883, A181647.

Select[Range[10000000], DivisorSigma[1, #]==DivisorSigma[1, # + 9] &] (* Vincenzo Librandi, Mar 10 2014 *)
Position[Partition[DivisorSigma[1,Range[10^7]],10,1],?(#[[1]] == #[[10]]&),{1},Heads->False]//Flatten (* _Harvey P. Dale, Aug 25 2016 *)

is(n)=sigma(n)==sigma(n+9) \\ Charles R Greathouse IV, Mar 09 2014

A015880 Numbers k such that sigma(k) = sigma(k+10).

Original entry on oeis.org

21, 174, 270, 517, 572, 913, 992, 1002, 1420, 1633, 1830, 2622, 2958, 4170, 4747, 5539, 7520, 7544, 7729, 10184, 10783, 14863, 16165, 16520, 19837, 20935, 21584, 23161, 26840, 28544, 29737, 31453, 34510, 35571, 35611, 35845, 39560

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000
A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

Cf. A002961, A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015881, A015882, A015883, A181647.

Select[Range[40000], DivisorSigma[1, #]==DivisorSigma[1, # + 10] &] (* Vincenzo Librandi, Mar 10 2014 *)

is(n)=sigma(n)==sigma(n+10) \\ Charles R Greathouse IV, Mar 09 2014

A015881 Numbers k such that sigma(k) = sigma(k+11).

Original entry on oeis.org

28, 154, 466, 874, 958, 1054, 2266, 2878, 11505, 12754, 14674, 17974, 21154, 21778, 29223, 29535, 31725, 32714, 39658, 43186, 48004, 52018, 62338, 70198, 126795, 132783, 163251, 164818, 207603, 212938, 221595, 272685, 274527

Offset: 1

Robert G. Wilson v

nonn

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

Cf. A002961, A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015882, A015883, A181647.

Select[Range[300000], DivisorSigma[1, #]==DivisorSigma[1, # + 11] &] (* Vincenzo Librandi, Mar 10 2014 *)
Position[Partition[DivisorSigma[1,Range[300000]],12,1],?(First[#]== Last[#]&), 1,Heads->False]//Flatten (* _Harvey P. Dale, Aug 20 2017 *)

is(n)=sigma(n)==sigma(n+11) \\ Charles R Greathouse IV, Mar 09 2014

cp's OEIS Frontend

A002961 Numbers k such that k and k+1 have same sum of divisors.

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A007373 Numbers k such that sigma(k+2) = sigma(k).

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A015861 Numbers k such that sigma(k) = sigma(k+3).

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A015865 Numbers k such that sigma(k) = sigma(k+5).

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A015863 Numbers k such that sigma(k) = sigma(k+4).

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A015866 Numbers k such that sigma(k) = sigma(k+6).

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A015876 Numbers k such that sigma(k) = sigma(k+8).

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A015877 Numbers k such that sigma(k) = sigma(k+9).

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