A002961 -id:A002961 - cp's OEIS frontend

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

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

A015882 Numbers k such that sigma(k) = sigma(k+12).

Original entry on oeis.org

35, 104, 285, 287, 310, 329, 340, 345, 406, 609, 660, 736, 767, 957, 1067, 1207, 1242, 1768, 1786, 1817, 1824, 2047, 2288, 2407, 2672, 2686, 2714, 3009, 4012, 4387, 4653, 4847, 6179, 7532, 8366, 8920, 10005, 10528, 11140, 11670, 11951

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, A015880, A015881, A015883, A181647.

Select[Range[20000], DivisorSigma[1, #]==DivisorSigma[1, # + 12] &] (* Vincenzo Librandi, Mar 10 2014 *)

is(n)=sigma(n)==sigma(n+12) \\ Charles R Greathouse IV, Mar 09 2014

A015883 Numbers k such that sigma(k) = sigma(k+13).

Original entry on oeis.org

182, 782, 1965, 2486, 2678, 2685, 12141, 12441, 17342, 21242, 27686, 34905, 35505, 35853, 38662, 38985, 56732, 63578, 104342, 109461, 192933, 198909, 222122, 236966, 245349, 251654, 256322, 261885, 262238, 324441, 333909

Offset: 1

Robert G. Wilson v

nonn

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 182, p. 56, Ellipses, Paris 2008.

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

Cf. A002961, A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A181647.

Select[Range[350000], DivisorSigma[1, #]==DivisorSigma[1, # + 13] &] (* Vincenzo Librandi, Mar 10 2014 *)

is(n)=sigma(n)==sigma(n+13) \\ Charles R Greathouse IV, Mar 09 2014

Corrected by T. D. Noe, Oct 31 2006

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

A181647 Numbers m having the same sum of divisors as m+20 has.

Original entry on oeis.org

42, 51, 123, 141, 204, 371, 497, 708, 923, 992, 1034, 1343, 1391, 1484, 1595, 1691, 1826, 3266, 3317, 5015, 5152, 7367, 8003, 9132, 9287, 9494, 11078, 13223, 15458, 15833, 17975, 18752, 19428, 20120, 20915, 21251, 21566, 24119, 24503, 25787, 28000, 29726, 29795

Offset: 1

Reinhard Zumkeller, Nov 03 2010

nonn

			a(1) = 42, divisors(42) = {1,2,3,6,7,14,21,42}, divisors(42+20) = {1,2,31,62}: 1+2+3+6+7+14+21+42 = 1+2+31+62.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 16.

Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 100 terms from Reinhard Zumkeller)
Andreas Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.5.

Cf. A000203, A002961, A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883.

Select[Range[30000], Equal @@ DivisorSigma[1, # + {0, 20}] &] (* Amiram Eldar, Apr 16 2025 *)

isok(n) = sigma(n) == sigma(n+20); \\ Michel Marcus, Feb 06 2016

A000203(a(n)) = A000203(a(n) + 20).

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

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

cp's OEIS Frontend

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

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

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A015882 Numbers k such that sigma(k) = sigma(k+12).

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A015883 Numbers k such that sigma(k) = sigma(k+13).

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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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A181647 Numbers m having the same sum of divisors as m+20 has.

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A340793 Sequence whose partial sums give A000203.