A055574 -id:A055574 - cp's OEIS frontend

A007373 Numbers k such that sigma(k+2) = sigma(k).

33, 54, 284, 366, 834, 848, 918, 1240, 1504, 2910, 2913, 3304, 4148, 4187, 6110, 6902, 7169, 7912, 9359, 10250, 10540, 12565, 15085, 17272, 17814, 19004, 19688, 21410, 21461, 24881, 25019, 26609, 28124, 30592, 30788, 31484, 38210, 38982, 39786, 40310, 45354

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

Numbers k such that antisigma(k+2) - antisigma(k) = 2*k + 3, where antisigma(m) = A024816(m) = sum of nondivisors of m. - Jaroslav Krizek, Mar 17 2013

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.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 33, pp 12, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)
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].
A. Weingartner, On the Solutions of sigma(n) = sigma(n+k), Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Essentially the same as A055574.

Cf. A002961, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647. [From Reinhard Zumkeller, Nov 03 2010]

Flatten[Position[Partition[DivisorSigma[1,Range[300000]],3,1], {x_, , x}]] (* Harvey P. Dale, Aug 08 2011 *)
SequencePosition[DivisorSigma[1,Range[300000]],{x_,,x}][[All,1]] (* Harvey P. Dale, Nov 17 2022 *)

je=[]; for(n=1,10^5,a=sigma(n); b=sigma(n+2); if(a==b,je=concat(je,n))); je

More terms from Jason Earls, Jul 20 2001

A067888 Numbers k such that tau(k+1) = tau(k-1) where tau(k) = A000005(k).

Original entry on oeis.org

4, 6, 7, 9, 12, 18, 19, 30, 34, 41, 42, 51, 55, 56, 60, 72, 86, 92, 94, 102, 103, 108, 124, 129, 137, 138, 142, 144, 150, 153, 160, 180, 183, 184, 185, 186, 192, 198, 199, 202, 204, 214, 216, 218, 220, 228, 231, 236, 240, 243, 244, 247, 248, 249, 266, 270, 282

Offset: 1

Benoit Cloitre, Mar 02 2002

If (p,p+2) are twin primes, then the composite number p+1 is in this sequence. The primes occurring in this sequence are listed in A067889. See A055574 for the analog with sigma instead of tau. - M. F. Hasler, Aug 06 2015

Giovanni Resta, Table of n, a(n) for n = 1..10000

Equals A062832 + 1. - Michel Marcus, Feb 11 2018

Cf. A000005, A055574, A067889.

Select[Range[300], Equal @@ DivisorSigma[0, # + {-1, 1}] &] (* Amiram Eldar, Jan 23 2025 *)

is_A067888(n)=n>1&&numdiv(n-1)==numdiv(n+1) \\ M. F. Hasler, Aug 06 2015

A226361 Numbers n such that sigma(n) = sigma(n+1) + sigma(n+2).

Original entry on oeis.org

378624, 661152, 5479092, 5526024, 7179624, 18744216, 122321970, 168201288, 215676636, 778701984, 1482154170, 1788138780, 1974360132, 2288979096, 3361923780, 4214315484, 4757106144, 4971510492, 6264306144, 6884356716, 10730488296, 11375549304, 16851779736

Offset: 1

Alex Ratushnyak, Jun 05 2013

nonn

Giovanni Resta, Table of n, a(n) for n = 1..88 (terms < 10^13)

Cf. A000203, A055574, A065900, A073500, A099632.

nn = 10^7; t = {}; sig0 = 1; sig1 = 3; Do[sig2 = DivisorSigma[1, n + 2]; If[sig0 == sig1 + sig2, AppendTo[t, n]]; sig0 = sig1; sig1 = sig2, {n, nn}]; t (* T. D. Noe, Jun 05 2013 *)

a(17)-a(23) from Donovan Johnson, Jun 05 2013

A067130 Numbers k such that sigma(k-1) divides sigma(k+1).

Original entry on oeis.org

2, 34, 55, 119, 285, 367, 835, 849, 919, 1241, 1505, 1559, 2911, 2914, 2939, 3305, 4149, 4188, 6111, 6903, 7170, 7913, 9360, 10251, 10541, 12566, 15086, 17273, 17759, 17815, 19005, 19689, 19919, 21411, 21462, 24882, 25020, 26610, 28125, 30593, 30789, 31485

Offset: 1

Benoit Cloitre, Feb 18 2002

nonn

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

Cf. A000203 (sigma), A055574.

Flatten[Position[Partition[DivisorSigma[1,Range[32000]],3,1],?( Divisible[ Last[#],First[#]]&),{1},Heads->False]]+1 (* _Harvey P. Dale, Jan 13 2015 *)

isok(n) = sigma(n+1) % sigma(n-1) == 0; \\ Michel Marcus, Nov 20 2013

More terms from Michel Marcus, Nov 20 2013

A226475 Numbers n such that sigma(n) + sigma(n+1) = sigma(n+2) + sigma(n+3).

Original entry on oeis.org

75, 113, 295, 533, 686, 2130, 14805, 26966, 30235, 35095, 135653, 355675, 432996, 590138, 1214588, 2692853, 2952064, 3375195, 3486795, 5973014, 6880351, 7334956, 22266602, 25841659, 30483834, 37416582, 38390010, 40952513, 41109593, 57242145

Offset: 1

Alex Ratushnyak, Jun 11 2013

nonn

Sigma(n) is the sum of the divisors of n: A000203.

			sigma(75) + sigma(76) = 124 + 140 = 264, and sigma(77) + sigma(78) = 96 + 168 = 264, so 75 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..138 (terms < 10^13)

Cf. A000203, A055574, A065900, A073500, A099632, A226361.

t = {}; s = DivisorSigma[1, Range[0, 3]]; n = 3; While[Length[t] < 10, n++; s = RotateLeft[s]; s[[4]] = DivisorSigma[1, n]; If[s[[1]] + s[[2]] == s[[3]] + s[[4]], AppendTo[t, n - 3]]]; t (* T. D. Noe, Jun 12 2013 *)
Module[{ds=DivisorSigma[1,Range[6*10^7]]},Flatten[Position[Partition[ds,4,1],?(Total[Take[#,2]]==Total[Take[#,-2]]&),{1},Heads->False]]] (* _Harvey P. Dale, Sep 13 2014 *)

A067134 Numbers n such that sigma(n+1) = 2*sigma(n-1).

Original entry on oeis.org

119, 1559, 2939, 17759, 19919, 32219, 33839, 55964, 71039, 186779, 308039, 511499, 523775, 553499, 699359, 838214, 1048904, 1159379, 1328939, 1333247, 1700039, 2462687, 2703887, 2956079, 3115319, 3561095, 3764207, 3972695, 7625879, 7852919, 8048963

Offset: 1

Benoit Cloitre, Feb 18 2002

nonn

For each term given here, n+1 is divisible by 3, but that's not always true; n=12396999 is a counterexample.

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

Cf. A055574, A067135.

Flatten[Position[Partition[DivisorSigma[1,Range[8100000]],3,1],?(Last[#] == 2*First[#]&),{1},Heads->False]]+1 (* _Harvey P. Dale, Aug 19 2013 *)

s1=1; s2=3; for(n=2, 10^8, s3=sigma(n+1); if(s3==2*s1, print1(n ", ")); s1=s2; s2=s3) /* Donovan Johnson, Apr 06 2013 */

Edited by Dean Hickerson, Feb 20 2002

A223137 Numbers n such that sigma(n+1) - sigma(n-1) = k*n for some integer k, where sigma(n) = A000203 (sum of divisors of n).

Original entry on oeis.org

5, 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, 39787, 40311, 45355

Offset: 1

Jaroslav Krizek, May 01 2013

nonn

Supersequence of A055574 for k=0 (n satisfying sigma(n+1) = sigma(n-1)). For number 5 is k=1. Are there other such number for k=1 or k=-1?

Corresponding values of integers k: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,....

			Number 5 is in sequence because sigma(6) - sigma(4) = 12 - 7 = 5 = 1 * 5; k=1.

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

Cf. A055574, A000203.

Select[Range[100000], IntegerQ[(DivisorSigma[1, # + 1] - DivisorSigma[1, # - 1])/#] &] (* T. D. Noe, May 02 2013 *)

Extended by T. D. Noe, May 02 2013

A227304 Numbers k such that sigma(k+1) divides sigma(k-1).

Original entry on oeis.org

34, 55, 285, 367, 835, 849, 919, 1241, 1505, 2911, 2914, 3305, 4149, 4188, 6111, 6903, 7170, 7913, 8506, 9360, 10251, 10541, 12566, 15086, 17273, 17815, 19005, 19689, 21411, 21462, 24882, 25020, 25501, 26610, 28125, 30361, 30593, 30789, 31485, 37741, 38211, 38983, 39787

Offset: 1

Alex Ratushnyak, Jul 05 2013

nonn

The sequence consists mainly of terms of A055574 = { n | sigma(n+1) = sigma(n-1) }. - M. F. Hasler, Aug 06 2015

			Sigma(8507) = 8736 divides sigma(8505) = 17472 = 8736*2, so 8506 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..6000 (term 1..280 from Harvey P. Dale)

Cf. A000203, A055574, A067130, A058073.

Flatten[Position[Partition[DivisorSigma[1,Range[40000]],3,1], ?(Divisible[ #[[1]],#[[3]]]&),{1},Heads->False]]+1 (* _Harvey P. Dale, Jul 15 2014 *)

A076666 Numbers n such that sigma(n) + sigma(n+3) = sigma(n+1) + sigma(n+2).

Original entry on oeis.org

2012, 2096, 15892, 17888, 39916, 102784, 141008, 146227, 482144, 487865, 1321312, 1887008, 2749057, 3513881, 7141158, 16767172, 17503912, 28122834, 30534728, 37453779, 42140437, 60994100, 67777337, 78251933, 113091820, 113768920, 129868059, 199240914, 240859196, 302897372

Offset: 1

Joseph L. Pe, Oct 25 2002

nonn

Each term of the sequence marks the start of four consecutive sigma-values for which the sum of the means equals the sum of the extremes.

			sigma(2012) + sigma(2015) = 3528 + 2688 = 6216; sigma(2013) + sigma(2014) = 2976 + 3240 = 6216, so 2012 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..90 (terms < 4*10^12)

Cf. A000203, A055574, A065900, A073500, A099632, A226361, A226475.

Select[Range[10^5], DivisorSigma[1, # ] + DivisorSigma[1, # + 3] == DivisorSigma[1, # + 1] + DivisorSigma[1, # + 2] &]

a(6)-a(26) from Donovan Johnson, Feb 01 2009

a(27)-a(30) from Alex Ratushnyak, Jun 29 2013

A294173 Numbers k whose nearest neighbors have the same number of divisors, the same number of distinct prime factors, and the same sum of divisors.

Original entry on oeis.org

34, 55, 919, 1241, 4149, 4188, 7170, 12566, 15086, 24882, 25020, 26610, 51836, 53964, 59988, 77058, 143370, 150420, 167561, 170562, 205728, 215070, 220818, 418308, 564858, 731321, 907255, 910316, 986154, 1239870, 1569336, 1622914, 1841861, 1887240, 1979307, 2229012, 2262108

Offset: 1

Torlach Rush, Feb 10 2018

nonn

mu(k-1) = mu(k+1), where mu(k) = A008683(k), since k-1 and k+1 have the same number of distinct prime factors.
tau(k-1) = tau(k+1) = abs(phi(k-1) - phi(k+1)) iff abs(phi(k-1) - phi(k+1)) = 4, where phi(j) is A000010. When tau(j) = 4 omega(j) = 2 and phi(j), the product of two even numbers is divisible by 4.
For known elements:
- sigma(k +- 1) and tau(k +- 1) the greatest common divisor is 4.
- sigma(k +- 1) is divisible by tau(k +- 1).
- the digital root of sigma(k +- 1) is either 3 or 9.
- the prime signature of k +- 1 is the same (see question below).
The first prime terms are 919, 110495719, 2587274227, 3908452759, 4020447619, and 9314901619. - Giovanni Resta, Feb 12 2018
Are the prime signatures of k +- 1 always the same? - Andrey Zabolotskiy, Feb 14 2018

			34 is in the sequence because tau(33)=tau(35)=4, omega(33)=omega(35)=2, and sigma(33)=sigma(35)=48.
919 is in the sequence because tau(918)=tau(920)=16, omega(918)=omega(920)=3, and sigma(918)=sigma(920)=2160.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1478

Intersection of A067888, A088070, and A055574.

Cf. A229254, A008683, A000010.

Filtered([2..2000000],k->Sigma(k-1)=Sigma(k+1) and Number(FactorsInt(k-1))=Number(FactorsInt(k+1)) and Tau(k-1)=Tau(k+1)); # Muniru A Asiru, Feb 17 2018

with(numtheory):
select(k->sigma(k-1)=sigma(k+1) and mobius(k-1)=mobius(k+1) and tau(k-1)=tau(k+1), [$2..2000000]); # Muniru A Asiru, Feb 17 2018

1 + Position[Partition[Array[{DivisorSigma[0, #], DivisorSigma[1, #], PrimeOmega[#]} &, 10^6], 3, 1], ?(#[[1]] == +#[[-1]] &), {1}, Heads -> False][[All, 1]] (* _Michael De Vlieger, Feb 17 2018 *)

list(lim)=my(v=List(),k2=7,s2=sigma(k2),k1=8,s1=sigma(k1),s); forfactored(k=9,1+lim\1, s=sigma(k); if(s==s2 && numdiv(k)==numdiv(k2) && omega(k)==omega(k2), listput(v,k1[1])); k2=k1; k1=k; s2=s1; s1=s); Vec(v) \\ Charles R Greathouse IV, Feb 20 2018

cp's OEIS Frontend

A007373 Numbers k such that sigma(k+2) = sigma(k).

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A067888 Numbers k such that tau(k+1) = tau(k-1) where tau(k) = A000005(k).

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A226361 Numbers n such that sigma(n) = sigma(n+1) + sigma(n+2).

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A067130 Numbers k such that sigma(k-1) divides sigma(k+1).

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A226475 Numbers n such that sigma(n) + sigma(n+1) = sigma(n+2) + sigma(n+3).

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A067134 Numbers n such that sigma(n+1) = 2*sigma(n-1).

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A223137 Numbers n such that sigma(n+1) - sigma(n-1) = k*n for some integer k, where sigma(n) = A000203 (sum of divisors of n).

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A227304 Numbers k such that sigma(k+1) divides sigma(k-1).

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