A053215 -id:A053215 - 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

A053249 Number of divisors of n such that n and n+1 have the same sum of divisors.

Original entry on oeis.org

4, 4, 8, 8, 12, 8, 8, 4, 6, 12, 10, 4, 16, 12, 8, 8, 8, 12, 16, 8, 8, 16, 16, 16, 16, 8, 16, 8, 16, 4, 16, 16, 16, 12, 24, 12, 16, 8, 16, 16, 8, 16, 16, 12, 16, 16, 16, 16, 12, 12, 12, 16, 16, 40, 16, 16, 32, 12, 24, 32, 24, 16, 16, 24, 24, 4, 24, 16, 64, 24, 16, 8, 16, 16, 16, 24, 32, 32, 20, 16

Offset: 1

Asher Auel, Jan 11 2000

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, A000005, A002961, A053215, A054002, A054003.

[#Divisors(n):n in [1..4000000]| SumOfDivisors(n) eq SumOfDivisors(n+1)]; // Marius A. Burtea, Sep 07 2019

Reap[ Do[ If[ DivisorSigma[1, n] == DivisorSigma[1, n + 1], tau = DivisorSigma[0, n]; Print[{n, tau}]; Sow[tau]], {n, 1, 4*10^6}]][[2, 1]] (* Jean-François Alcover, Oct 08 2012 *)
DivisorSigma[0,#]&/@Flatten[Position[Partition[DivisorSigma[1,Range[ 4000000]],2,1], ?(First[#] == Last[#]&),{1},Heads->False]] (* _Harvey P. Dale, Jul 04 2014 *)
DivisorSigma[0,#]&/@(SequencePosition[DivisorSigma[1,Range[4000000]],{x_,x_}][[All,1]]) (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Jul 25 2019 *)

do(lim)=my(v=List(),k=1,t); for(n=2,lim, t=sigma(n); if(t==k, listput(v, numdiv(n-1))); k=t); Vec(v) \\ Charles R Greathouse IV, Feb 08 2017

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

More terms from Naohiro Nomoto, Mar 16 2001

A290303 Values of usigma(n) = usigma(n+1).

Original entry on oeis.org

24, 60, 72, 180, 1440, 2160, 1872, 2640, 2400, 3000, 2880, 3024, 4320, 4320, 4320, 5280, 5280, 7400, 8640, 10080, 10200, 11520, 11880, 11520, 11088, 12960, 12096, 14400, 25920, 21600, 26640, 34560, 25200, 40320, 34560, 36000, 51840, 60480, 63360, 60480, 65280

Offset: 1

Amiram Eldar, Jul 26 2017

nonn

The sum of unitary divisors of numbers n such that n and n+1 have the same sum.

The unitary version of A053215.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002961, A034448, A053215, A064125.

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]];a={}; u1=0; For[k=0, k<10^5, k++; u2=usigma[k]; If[u1==u2, a = AppendTo[a, u1]]; u1=u2]; a

a(n) = A034448(A064125(n)).

A294029 Values of bsigma(k) = bsigma(k+1), where bsigma is the sum of the bi-unitary divisors (A188999).

Original entry on oeis.org

24, 40, 60, 720, 960, 1440, 2160, 2640, 2400, 3000, 4320, 4320, 4320, 5280, 7400, 11520, 11880, 12960, 14400, 20160, 30240, 26640, 34560, 25200, 34560, 49920, 51840, 60480, 63360, 60480, 65280, 62400, 61560, 115200, 93600, 114912, 100800, 120960, 120960

Offset: 1

Amiram Eldar, Oct 22 2017

nonn

The sum of bi-unitary divisors of numbers n such that n and n+1 have the same sum (A293183).

The bi-unitary version of A053215.

			24 is in the sequence since 24 = bsigma(14) = bsigma(15).

Amiram Eldar, Table of n, a(n) for n = 1..2148

Cf. A053215, A290303, A188999, A293183.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; a = {}; b1 = 0; For[k = 0, k < 10^6, k++; b2 = bsigma[k]; If[b1 == b2, a = AppendTo[a, b1]]; b1 = b2]; a (* after Michael De Vlieger at A188999 *)

a(n) = A188999(A293183(n)).

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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A053249 Number of divisors of n such that n and n+1 have the same sum of divisors.

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A290303 Values of usigma(n) = usigma(n+1).

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A294029 Values of bsigma(k) = bsigma(k+1), where bsigma is the sum of the bi-unitary divisors (A188999).

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

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