A053249 -id:A053249 - 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

A054004 Numbers k such that k and k+1 have the same number and sum of divisors.

Original entry on oeis.org

14, 1334, 1634, 2685, 33998, 42818, 64665, 84134, 109214, 122073, 166934, 289454, 383594, 440013, 544334, 605985, 649154, 655005, 792855, 1642154, 2284814, 2305557, 2913105, 3571905, 3682622, 4701537, 5181045, 6431732

Offset: 1

Asher Auel, Jan 12 2000

nonn

			Divisors of 14 = {1, 2, 7, 14}, divisors of 15 = {1, 3, 5, 15}, both have four divisors and sum = 24.

Amiram Eldar, Table of n, a(n) for n = 1..1831 (calculated using the b-file at A002961; terms 1..967 from T. D. Noe)

Cf. A000005, A000203, A002961, A005237, A053249, A054005, A054006, A054007.

Select[Range[100000], DivisorSigma[0, #] == DivisorSigma[0, # + 1] && DivisorSigma[1, #] == DivisorSigma[1, # + 1] &] (* Jayanta Basu, Mar 20 2013 *)

More terms from Jud McCranie, Jun 02 2000

A053215 Sum of divisors of those numbers n such that n and n+1 have the same sum of divisors.

Original entry on oeis.org

24, 312, 1440, 2160, 2688, 2640, 4320, 4464, 7644, 22932, 28314, 29040, 34560, 37440, 51840, 56160, 65280, 100800, 115200, 114912, 120960, 120960, 138240, 153216, 194400, 168960, 178560, 186048, 207360, 221184, 244800, 280800, 276480

Offset: 1

Asher Auel, Jan 11 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10135 (from the b-file at A002961; terms 1..4804 from T. D. Noe)

Cf. A000203, A002961, A053249.

Transpose[Select[{DivisorSigma[1,First[#]],DivisorSigma[1,Last[#]]}&/@ Partition[Range[280000],2,1],First[#]==Last[#]&]][[1]] (* Harvey P. Dale, May 07 2012 *)
DivisorSigma[1,#]&/@SequencePosition[DivisorSigma[1,Range[280000]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 21 2017 *)

a(n) = sigma(A002961(n))

A054005 Sum of divisors of k such that k and k+1 have the same number and sum of divisors.

Original entry on oeis.org

24, 2160, 2640, 4320, 51840, 65280, 115200, 138240, 194400, 186048, 276480, 483840, 622080, 700416, 950400, 984960, 1118880, 1128960, 1612800, 2661120, 3937248, 3617280, 5019840, 6128640, 5806080, 7375680, 8467200, 11583936

Offset: 1

Asher Auel, Jan 12 2000

nonn

			See example in A054004.

Amiram Eldar, Table of n, a(n) for n = 1..1831

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054006, A054007.

Select[Partition[Table[{n,DivisorSigma[0,n],DivisorSigma[1,n]},{n,116*10^5}],2,1],#[[1,2]]== #[[2,2]] && #[[1,3]]==#[[2,3]]&][[All,1,3]] (* Harvey P. Dale, May 16 2023 *)

a(n) = sigma(A054004(n)).

More terms from Jud McCranie, Oct 15 2000

Definition clarified by Harvey P. Dale, May 16 2023

A054006 Number of divisors of k and k+1 which have the same number and sum of divisors.

Original entry on oeis.org

4, 8, 8, 8, 8, 8, 16, 16, 16, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 16, 16, 16, 16, 32, 16, 16, 16, 24, 16, 16, 16, 32, 32, 32, 16, 16, 32, 16, 32, 16, 16, 32, 32, 16, 32, 16, 16, 16, 16, 16, 32, 32, 16, 32, 16, 16, 64, 32, 16, 32, 16, 32, 16, 64, 32, 32, 16, 32, 32, 32, 32

Offset: 1

Asher Auel, Jan 12 2000

nonn

			See example in A054004.

Amiram Eldar, Table of n, a(n) for n = 1..1831 (calculated using the b-file at A054004)

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054005, A054007.

Select[Partition[Array[DivisorSigma[{0, 1}, #] &, 10^6], 2, 1], SameQ @@ # &][[All, 1, 1]] (* Michael De Vlieger, Nov 21 2019 *)

a(n) = tau(A054004(n)).

More terms from Jud McCranie, Oct 15 2000

A054007 Numbers k such that k and k+1 have the same sum but an unequal number of divisors.

Original entry on oeis.org

206, 957, 1364, 2974, 4364, 14841, 18873, 19358, 20145, 24957, 36566, 56564, 74918, 79826, 79833, 92685, 111506, 116937, 138237, 147454, 161001, 162602, 174717, 190773, 193893, 201597, 230390, 274533, 347738, 416577, 422073, 430137

Offset: 1

Asher Auel, Jan 12 2000

nonn

			The divisors of 206 are 1, 2, 103, 206, so tau(206) = 4 and sigma(206) = 312; the divisors of 207 are 1, 3, 9, 23, 69, 207, so tau(207) = 6 and sigma(207) = 312. Hence, the integer 206 belongs to this sequence. - _Bernard Schott_, Oct 18 2019

Amiram Eldar, Table of n, a(n) for n = 1..8304 (terms below 10^13, calculated from the b-file at A002961)

Cf. A000005, A000203, A002961, A005237, A053249, A054004, A054005, A054006.

Select[Range[100000], DivisorSigma[0, #] != DivisorSigma[0, # + 1] && DivisorSigma[1, #] == DivisorSigma[1, # + 1] &] (* Jayanta Basu, Mar 20 2013 *)

Members of A002961 which are not members of A054004

More terms from Jud McCranie, Oct 15 2000

A054002 Number of divisors of n such that n and n-1 have the same sum of divisors.

Original entry on oeis.org

4, 6, 4, 8, 16, 8, 8, 12, 12, 8, 4, 10, 8, 4, 8, 12, 8, 16, 16, 16, 16, 8, 16, 12, 16, 16, 8, 8, 4, 16, 8, 24, 16, 4, 16, 8, 4, 32, 8, 16, 16, 16, 8, 8, 8, 8, 16, 32, 8, 8, 16, 16, 16, 12, 16, 16, 12, 8, 48, 32, 8, 24, 24, 16, 16, 16, 8, 16, 24, 64, 8, 16, 16, 16, 16, 64, 24, 32, 60

Offset: 1

Asher Auel, Jan 12 2000

nonn

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

Cf. A000005, A000203, A002961, A053249, A054003.

[#Divisors(n):n in [2..3500000]| SumOfDivisors(n) eq SumOfDivisors(n-1)]; // Marius A. Burtea, Sep 07 2019

a(n) = tau(A002961(n) + 1).

More terms from Naohiro Nomoto, Jun 23 2001

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

Original entry on oeis.org

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

A372114 Sum of squares of divisors of the numbers m such that m and m+2 have the same sum of squares of divisors.

Original entry on oeis.org

850, 48100, 110500, 1171300, 897826072900, 1855703820100, 4974132151300, 223203708201604, 388880538297700, 1056863959716100, 2077699792101700, 2101425630304900, 2444010061663300, 6152287246125700, 6862948725741700, 10203957350659300, 27831593408440900, 50905357902220900

Offset: 1

Amiram Eldar, Apr 19 2024

nonn

All the terms are even.

There are only 2 equal consecutive terms in A001157: sigma_2(6) = sigma_2(7) = 50.

Amiram Eldar, Table of n, a(n) for n = 1..156
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.

Cf. A001157, A169635.

Similar sequences: A053215, A053249.

seq[mmax_] := Module[{s1 = DivisorSigma[2, 1], s2 = DivisorSigma[2, 2], s3, s4, s={}}, Do[s3 = DivisorSigma[2, m]; s4 = DivisorSigma[2, m+1]; If[s1 == s3, AppendTo[s, s1]]; If[s2 == s4, AppendTo[s, s2]]; s1 = s3; s2 = s4, {m, 3, mmax, 2}]; s]; seq[10^6]

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(s1, ", ")); if(s2 == s4, print1(s2, ", ")); s1 = s3; s2 = s4);}

a(n) = A001157(A169635(n)).

cp's OEIS Frontend

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

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A054004 Numbers k such that k and k+1 have the same number and sum of divisors.

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A053215 Sum of divisors of those numbers n such that n and n+1 have the same sum of divisors.

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A054005 Sum of divisors of k such that k and k+1 have the same number and sum of divisors.

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A054006 Number of divisors of k and k+1 which have the same number and sum of divisors.

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A054007 Numbers k such that k and k+1 have the same sum but an unequal number of divisors.

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A054002 Number of divisors of n such that n and n-1 have the same sum of divisors.

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

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