A015861 -id:A015861 - cp's OEIS frontend

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

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

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).

A276714 Numbers n such that n and n+3 have the same number and sum of divisors (A000005 and A000203).

Original entry on oeis.org

42677635, 276742235, 6439057062, 7512673242, 43592652562, 48847956255, 48880963215, 55018687182, 60184185702, 91484515395, 100774916235, 101926379835, 111886551315, 122388340095, 133012188855, 137978601142, 247631352255, 263171068875, 293467635615, 305946896255

Offset: 1

Jaroslav Krizek, Sep 16 2016

nonn

Intersection of A015861 and A276713.

Also numbers n such that A229335(n) = A229335(n+3).

			42677635 is in sequence because tau(42677635) = tau(42677638) = 32 and sigma(42677635) = sigma(42677638) = 68769792.

Martin Ehrenstein, Table of n, a(n) for n = 1..30

Cf. A015861, A229335, A276713, A276715.

Cf. Similar sequences with numbers n such that n and n+k have the same number and sum of divisors for k=1: A054004, for k=2: A229254.

[n: n in [A015861(k)] | NumberOfDivisors(n) eq  NumberOfDivisors(n+3) and SumOfDivisors(n) eq  SumOfDivisors(n+3)]

More terms from Martin Ehrenstein, Jul 12 2024

A172333 Numbers m such that m and m+22 have the same sum of divisors.

Original entry on oeis.org

57, 85, 213, 224, 354, 476, 568, 594, 812, 1218, 1235, 1316, 1484, 2103, 2470, 2492, 2643, 2840, 2996, 3836, 3978, 4026, 4544, 4810, 4844, 5012, 6125, 6356, 6524, 7364, 7532, 7648, 8876, 9272, 9328, 10098, 11107, 11797, 12572, 12594, 13412, 13640

Offset: 1

Michel Lagneau, Feb 01 2010

nonn

If 3*k-1 and 14*k-1 are both prime with k>1, then n = 28*(3*k-1) belongs to this sequence. The number of such integers n <= x would be asymptotically cx/(log x)^2 for some constant c > 0 from the Hardy-Littlewood conjecture D in Partitio Numerorum. - Tomohiro Yamada, Oct 03 2018

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 62, p. 22, Ellipses, Paris 2008.
W. Sierpinski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
Tomohiro Yamada, On equations sigma(n) = sigma(n+k) and phi(n) = phi(n+k), J. Comb. Number Theory 9 (2017), 15-21.

Tomohiro Yamada, Table of n, a(n) for n = 1..46702 (All terms < 2^28, first 2000 terms from Muniru A Asiru)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.
Tomohiro Yamada, On equations sigma(n) = sigma(n+k) and phi(n) = phi(n+k)<, arXiv:1001.2511 [math.NT], 2010.

Cf. A000203, A015861, A002961, A015865, A015867, A015858, A015859, A015860.

Filtered([1..13700],k->Sigma(k)=Sigma(k+22)); # Muniru A Asiru, Oct 20 2018

with(numtheory):for n from 1 to 20000 do;if sigma(n) = sigma(n+22) then print(n); else fi ; od;

isok(k) = sigma(k)==sigma(k+22); \\ Altug Alkan, Oct 03 2018

A276713 Numbers n such that n and n+3 have the same number of divisors (A000005).

Original entry on oeis.org

2, 35, 55, 62, 74, 82, 91, 102, 115, 119, 122, 135, 142, 143, 155, 158, 172, 186, 202, 203, 206, 214, 215, 218, 242, 255, 259, 262, 282, 295, 298, 299, 302, 323, 326, 343, 351, 354, 355, 362, 391, 395, 399, 425, 426, 435, 451, 466, 478, 482, 492, 502, 511, 514

Offset: 1

Jaroslav Krizek, Sep 16 2016

nonn

			35 is in sequence because tau(35) = tau(38) = 4.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A065559 (smallest k such that tau(k) = tau(k+n)), A015861 (sigma(n) = sigma(n+3)), A276714.

Cf. Similar sequences with numbers n such that n and n+k have the same number of divisors for k = 1: A005237, for k = 2: A062832.

[n: n in [1..10000] | NumberOfDivisors(n) eq  NumberOfDivisors(n+3)]

with(numtheory): A276713:=n->`if`(tau(n) = tau(n+3), n, NULL): seq(A276713(n), n=1..10^3); # Wesley Ivan Hurt, May 02 2017

SequencePosition[DivisorSigma[0,Range[600]],{x_,,,x_}][[All,1]] (* Harvey P. Dale, Nov 12 2022 *)

isok(n) = numdiv(n) == numdiv(n+3); \\ Michel Marcus, May 03 2017

A321023 Numbers k such that sigma(k) = sigma(k + 15).

Original entry on oeis.org

26, 62, 20840574, 25741470, 60765690, 102435795, 277471467, 361466454, 464465910, 1110512403, 1927430490, 2741174163, 3631266639, 3844534602, 3982743750, 4565968407, 4612184562, 4829319495, 4981969978, 7066794735, 13484870399, 14268004443, 14550390855, 15051147111

Offset: 1

Tomohiro Yamada, Oct 26 2018

nonn

			sigma(26) = sigma(41) = 42, sigma(62) = sigma(77) = 96.

Giovanni Resta, Table of n, a(n) for n = 1..116 (terms < 2.5*10^12)

Cf. A002961, A007373, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A172335, A181647, A172333.

my(V=vector(15)); for(n=1, 2^28, my(s=sigma(n), r=(n%15)+1); if (s==V[r], print1(n-15, ", ")); V[r]=s)

a(7)-a(9) from Amiram Eldar, Oct 26 2018

a(10)-a(24) from Giovanni Resta, Oct 26 2018

cp's OEIS Frontend

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

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A276714 Numbers n such that n and n+3 have the same number and sum of divisors (A000005 and A000203).

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A172333 Numbers m such that m and m+22 have the same sum of divisors.

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A276713 Numbers n such that n and n+3 have the same number of divisors (A000005).

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A321023 Numbers k such that sigma(k) = sigma(k + 15).

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