A007373 -id:A007373 - 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).

A055574 n satisfying sigma(n+1) = sigma(n-1).

Original entry on oeis.org

34, 55, 285, 367, 835, 849, 919, 1241, 1505, 2911, 2914, 3305, 4149, 4188, 6111, 6903, 7170, 7913, 9360, 10251, 10541, 12566, 15086, 17273, 17815, 19005, 19689, 21411, 21462, 24882, 25020, 26610, 28125, 30593, 30789, 31485, 38211, 38983

Offset: 1

Joseph L. Pe, Feb 12 2002

nonn

Essentially the same as A007373: a(n) = A007373(n) + 1.

Numbers n such that antisigma(n+1) - antisigma(n-1) = 2*n + 1, where antisigma(m) = A024816(m) = sum of nondivisors of m. - Jaroslav Krizek, Mar 17 2013

			sigma(34-1) = 48 = sigma(34+1), so 34 is a term of the sequence.

Vincenzo Librandi and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 200 terms from Vincenzo Librandi)

Cf. A007373.

Select[Range[10^5], DivisorSigma[1, # + 1] == DivisorSigma[1, # - 1] &]

is(n)=sigma(n+1)==sigma(n-1) \\ Charles R Greathouse IV, Mar 09 2014

x=y=1; forfactored(z=3,10^6, if(sigma(z)==sigma(x), print1(y[1]", ")); x=y; y=z) \\ Charles R Greathouse IV, May 09 2017

A229254 Numbers k such that k and k+2 have the same number (A000005) and sum of divisors (A000203).

Original entry on oeis.org

33, 54, 918, 1240, 3304, 4148, 4187, 7169, 12565, 15085, 19688, 24881, 25019, 26609, 38982, 51835, 53963, 59987, 76360, 77057, 96728, 143369, 150419, 167560, 170561, 205727, 215069, 220817, 278920, 418307, 564857, 731320, 785270, 907254, 910315, 986153

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

Also numbers k such that A229335(k) = A229335(k+2).

Intersection of A007373 and A062832.

			Divisors of 54 = {1, 2, 3, 6, 9, 18, 27, 54}, divisors of 56 = {1, 2, 4, 7, 8, 14, 28, 56}, both have 8 divisors and sum = 120.

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

Cf. A007373, A062832, A000005, A000203, A229335.

Select[Range[10000], DivisorSigma[0, #] == DivisorSigma[0, # + 2] && DivisorSigma[1, #] == DivisorSigma[1, # + 2] &]

isok(n) = (numdiv(n) == numdiv(n+2)) && (sigma(n) == sigma(n+2)); \\ Michel Marcus, Sep 20 2013

More terms from Michel Marcus, Sep 20 2013

A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).

Original entry on oeis.org

24, 215, 280, 1079, 947519, 1362239, 2230271, 14939999, 19720007, 32509439, 45581759, 45841247, 49436927, 78436511, 82842911, 101014631, 166828031, 225622151, 225757799, 250999559, 377129087, 554998751, 619606439, 846765431, 1204092287, 1302170687, 1710035711

Offset: 1

Michel Lagneau, Apr 04 2010

nonn

The equation sigma_2(m) = sigma_2(m + k) has infinitely many solutions where k >= 2 and k is even (J.-M. De Koninck).
From Amiram Eldar, Apr 19 2024: (Start)
De Koninck's proof is based on the assumption of Schinzel's hypothesis H. If q, r = q + 2, s = q^2 + q + 1, and p = q^2 + 3*q + 3 are all primes, then p*q is a term (the values of q+1 are the terms of A268043).
The equation sigma_2(m) = sigma_2(m + 1) has only one solution: m = 6. (End)

			For m=24, sigma_2(24) = sigma_2(26) = 850.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 118, entry 1079.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B13, pp. 103-104.

Amiram Eldar, Table of n, a(n) for n = 1..156
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].
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.
Wikipedia, Schinzel's hypothesis H.

Cf. A001157, A002961, A007373, A175199, A268043.

with(numtheory):for n from 1 to 500000000 do:liste:= divisors(n) : s2 :=sum(liste[i]^2, i=1..nops(liste)):liste:=divisors(n+2):s3:=sum(liste[i]^2, i=1..nops(liste)):if s2 = s3 then print(n):else fi:od:

Select[Range[10^9], DivisorSigma[2,#] == DivisorSigma[2,#+2]&]

is(n) = sigma(n, 2) == sigma(n + 2, 2); \\ Amiram Eldar, Apr 19 2024

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(m - 2, ", ")); if(s2 == s4, print1(m - 1, ", ")); s1 = s3; s2 = s4);} \\ Amiram Eldar, Apr 19 2024

a(25)-a(27) from Donovan Johnson, Apr 14 2013

A330703 Numbers k such that psi(k) = psi(k + 2) where psi(k) is the Dedekind psi function (A001615).

Original entry on oeis.org

6, 9, 12, 14, 18, 20, 33, 44, 62, 70, 92, 108, 116, 138, 164, 175, 212, 254, 280, 308, 320, 332, 348, 356, 452, 490, 524, 558, 572, 692, 716, 764, 833, 932, 956, 1004, 1105, 1124, 1172, 1188, 1436, 1496, 1562, 1593, 1676, 1724, 1772, 1964, 2002, 2036, 2088, 2132

Offset: 1

Amiram Eldar, Dec 26 2019

nonn

			6 is in the sequence since psi(6) = psi(8) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jozsef Sandor, On the composition of some arithmetic functions, II, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, No. 3 (2005), Article 73.

Cf. A001494, A001615, A007373, A062832, A291043.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[10^3], psi[#] == psi[# + 2] &]

A283552 Numbers k == 33 (mod 60) such that 2k+1, 2k+5, 3k+2 and 3k+8 are all primes.

Original entry on oeis.org

33, 153, 453, 1953, 4773, 19353, 23253, 36273, 37413, 38793, 40773, 50133, 51693, 70413, 70833, 83433, 88893, 108393, 115233, 117873, 131193, 136113, 157773, 161733, 164793, 170973, 184533, 221793, 234813, 238293, 258453, 271893, 272313, 287313, 304953, 307713, 325533, 327753, 330393

Offset: 1

Amiram Eldar, Mar 10 2017

nonn

Andreas Weingartner used the first 913685 terms of this sequence to prove that the equation sigma(x) = sigma(x+k) has at least one solution for every even k in the range 2 <= k <= 10^(10^7). The upper bound is just lower than the product of 2a(n)+1 of these terms which equals 3.222... * 10^10000007.

			a(2) = 153, 2*153 + 1 = 307, 2*153 + 5 = 311, 3*153 + 2 = 461 and 3*153 + 8 = 467 are all primes.

Amiram Eldar, 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. A000203, A007373.

Select[33 + Range[0, 6*10^5]*60, PrimeQ[2 # + 1] && PrimeQ[2 # + 5] && PrimeQ[3 # + 2] && PrimeQ[3 # + 8] &]

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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A055574 n satisfying sigma(n+1) = sigma(n-1).

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A229254 Numbers k such that k and k+2 have the same number (A000005) and sum of divisors (A000203).

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A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).

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A330703 Numbers k such that psi(k) = psi(k + 2) where psi(k) is the Dedekind psi function (A001615).

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