A015915 -id:A015915 - cp's OEIS frontend

A023202 Primes p such that p + 8 is also prime.

3, 5, 11, 23, 29, 53, 59, 71, 89, 101, 131, 149, 173, 191, 233, 263, 269, 359, 389, 401, 431, 449, 479, 491, 563, 569, 593, 599, 653, 683, 701, 719, 743, 761, 821, 911, 929, 983, 1013, 1031, 1061, 1109, 1163, 1193, 1223, 1229, 1283, 1289, 1319, 1373, 1439

Offset: 1

David W. Wilson

All terms > 3 are congruent to 5 mod 6 (observation by Zak Seidov in SeqFan). Thus each corresponding p + 8 is congruent to 1 mod 6. - Rick L. Shepherd, Mar 25 2023

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, corrected by Sean A. Irvine and Georg Fischer)
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Twin Primes

Disjoint union of A007530, A031926, A049437, A049438.

Cf. A046134, A049436, A046138, A015915.

Filtered([1..1500], k-> IsPrime(k) and IsPrime(k+8)); # G. C. Greubel, Feb 07 2020

[n: n in [0..1500] | IsPrime(n) and IsPrime(n+8)]; // Vincenzo Librandi, Nov 20 2010

select(n-> isprime(n) and isprime(n+8), [`$`(1..1500)]); # G. C. Greubel, Feb 07 2020

Select[Range[1500], PrimeQ[#] && PrimeQ[#+8]&] (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[250]],PrimeQ[#+8]&] (* Harvey P. Dale, Dec 24 2020 *)

is(n)=isprime(n)&&isprime(n+8) \\ Charles R Greathouse IV, Jul 01 2013

[n for n in (1..1500) if is_prime(n) and is_prime(n+8)] # G. C. Greubel, Feb 07 2020

A054905 Smallest composite x such that sigma(x) + 2n = sigma(x + 2n).

Original entry on oeis.org

434, 305635357, 104, 27, 195556, 65, 12, 39, 20, 56, 916, 80, 212282, 57, 44, 106645, 52, 125

Offset: 1

Labos Elemer May 23 2000

a(19) > 4293000000, if it exists. - Jud McCranie, May 25 2000

a(19) > 10^11, if it exists. - Charles R Greathouse IV, Oct 26 2022

			a(5) corresponds to n=3+2=5, d=2n=10 and the smallest composite integer is 195556. The next solution is 1152136225.

Mauro Fiorentini, Le soluzioni dell’equazione s(n + k) = s(n) + k, con n composto fino a 109 e k sino a 1000.

Cf. A015913, A015914, A015915, A015916, A015917, A054799.

Cf. A023200, A023201, A023202, A023203,

a(n)=forcomposite(x=3,10^66,if(sigma(x)+2*n==sigma(x+2*n),return(x)));
for(n=1,66,print1(a(n),", ")); \\ Joerg Arndt, Nov 15 2014

a19(lim,startAt=39)=startAt=ceil(startAt); my(v=vectorsmall(38),i=(startAt-1)%38); forfactored(n=startAt,lim\1+38, my(t=sigma(n)); if(i++>38,i=1); if(t==v[i]+38, return(n[1]-38)); v[i]=if(vecsum(n[2][,2])>1,t,0)) \\ Charles R Greathouse IV, Oct 25 2022

Description corrected by Jud McCranie, May 25 2000

A015916 Numbers k such that sigma(k) + 10 = sigma(k+10).

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 73, 79, 97, 103, 127, 139, 157, 163, 181, 223, 229, 241, 271, 283, 307, 337, 349, 373, 379, 409, 421, 433, 439, 457, 499, 547, 577, 607, 631, 643, 673, 691, 709, 733, 751, 787, 811, 829, 853, 877, 919, 937, 967, 1009

Offset: 1

Robert G. Wilson v

nonn

Different from A023203. Below 1000000 the only composite number here is 195556: sigma(195556) + 10 = 342230 + 10 = sigma(195566). - Labos Elemer, May 23 2000

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A023203, A015913, A015914, A015915, A015917.

Select[Range[2000], DivisorSigma[1, #] + 10==DivisorSigma[1, # + 10] &] (* Vincenzo Librandi, Mar 10 2014 *)
Select[Partition[DivisorSigma[1,Range[1100]],11,1],#[[1]]+10==#[[-1]]&][[All,1]]-1 (* Harvey P. Dale, May 20 2021 *)

A054987 Smallest composite x such that sigma(x+2^n) = sigma(x) + 2^n.

Original entry on oeis.org

434, 305635357, 27, 39, 106645, 69, 2275, 63, 6475, 249, 7735, 3703, 10803, 16383, 58869, 51181, 87951, 1695, 9579, 105237, 98829, 1143369, 789609, 11625, 14038691, 178975, 48627929, 1881333, 402373721, 2667945, 82915599, 353195221, 70106601

Offset: 1

Labos Elemer, May 29 2000

nonn

The sequence is initiated by smallest of A015915. Special primes of A023202, A049488-A049491 also satisfy the Sigma[p+2^w]=Sigma[p]+2^w relation

			For the term 69: Sigma[69+2^6] = Sigma[133] = 1+7+19+133 = Sigma[69]+64 = (1+3+23+69)+64 = 160.

Cf. A049488-A049491, A001359, A054799, A015913, A015915, A023200, A023202, A054905.

Table[ Select[ Range[ 1, 110000 ], Equal[ EulerPhi[ #+2^k ]-EulerPhi[ # ]-2^k, 0 ] &&!PrimeQ[ # ]& ], {k, 1, 22} ]

a(n)=my(N=2^n,x=3); while(isprime(x++) || sigma(x+N) != sigma(x)+N,); x \\ Charles R Greathouse IV, Mar 11 2014

More terms from Labos Elemer, Aug 14 2003

a(21) corrected and a(27)-a(33) from Donovan Johnson, Nov 30 2008

A059118 Composite solutions to sigma(x)+8 = sigma(x+8).

Original entry on oeis.org

27, 1615, 1885, 218984, 4218475, 312016315, 746314601, 1125845307, 1132343549, 1296114929, 9016730984, 303419868239, 1197056419121, 2065971192041, 2948269852109, 4562970154601

Offset: 1

Jud McCranie, Jan 06 2001

a(17) > 10^13. - Giovanni Resta, Jul 11 2013

			sigma(27)+8 = 48 = sigma(27+8), so 27 is in the sequence.

The first 4 terms were found by Labos Elemer, see A015915.

Cf. A015915.

ta={{0}};Do[If[Equal[DivisorSigma[1, n+8] -DivisorSigma[1, n]-8, 0]&&!PrimeQ[n], Print[n]; ta=Append[ta, n]], {n, 1000000000, 1300000000}]; ta=Delete[ta, 1] (* Labos Elemer, Jan 10 2005 *)

a(8)-a(10) terms from Labos Elemer, Jan 10 2005
Offset corrected and a(11) added by Donovan Johnson, Dec 07 2008
a(12)-a(16) from Giovanni Resta, Jul 11 2013

A054984 Composite numbers k such that sigma(k + 6!) = sigma(k + 720) = sigma(k) + 720.

Original entry on oeis.org

427, 553, 595, 623, 737, 871, 913, 923, 1199, 1207, 1241, 1507, 1582, 1817, 1848, 2193, 2226, 2337, 2398, 2407, 2553, 2561, 2728, 2758, 2929, 3016, 3115, 3248, 3346, 3502, 3503, 3598, 3705, 3762, 4171, 4293, 4343, 4462, 4587, 4633, 4841, 4867, 4984

Offset: 1

Labos Elemer, May 29 2000

nonn

			553 is a term because sigma(553) + 720 = 640 + 720 = 1360 = sigma(553 + 720) = sigma(1273) = 1 + 19 + 67 + 1273.

Amiram Eldar, Table of n, a(n) for n = 1..6000

Cf. A015914, A015915, A015916, A015917, A033560, A033561, A054799.

Select[Range[5000], CompositeQ[#] && Differences@ DivisorSigma[1, {#, # + 720}] == {720} &] (* Amiram Eldar, Mar 09 2025 *)

isok(k) = !isprime(k) && sigma(k + 720) == sigma(k) + 720; \\ Amiram Eldar, Mar 09 2025

A063680 Solutions to sigma(k) + 7 = sigma(k+7).

Original entry on oeis.org

74, 531434, 387420482, 2541865828322

Offset: 1

Jud McCranie, Jul 28 2001

No other solutions < 4290000000. Sequence A063679 shows how to generate more solutions, but there may be solutions other than those produced by A063679.
No others < 10^17. - Seth A. Troisi, Oct 25 2022
k or k+7 must be a square or twice a square (A028982). See comment in A015886. - Seth A. Troisi, Oct 26 2022
From Jon E. Schoenfield, Oct 26 2022: (Start)
Each of the first 4 terms of the sequence is of the form k = 9^j - 7:
             74 = 9^2  - 7,
         531434 = 9^6  - 7,
      387420482 = 9^9  - 7,
  2541865828322 = 9^13 - 7.
The next terms of this form are 9^53 - 7 and 9^82 - 7.
Does the sequence contain any terms that are not of this form?
(End)
No other terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			sigma(74) + 7 = 121 = sigma(74+7), so 74 is in the sequence.

Cf. A000203, A063679.

Cf. A054799, A015913, A015914, A015915, A015916, A015917, A054905.

isok(k) = sigma(k) + 7 == sigma(k+7); \\ Michel Marcus, Oct 25 2022

a(4) from Seth A. Troisi, Oct 24 2022

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A023202 Primes p such that p + 8 is also prime.

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A054905 Smallest composite x such that sigma(x) + 2n = sigma(x + 2n).

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A015916 Numbers k such that sigma(k) + 10 = sigma(k+10).

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A054987 Smallest composite x such that sigma(x+2^n) = sigma(x) + 2^n.

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A059118 Composite solutions to sigma(x)+8 = sigma(x+8).

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A054984 Composite numbers k such that sigma(k + 6!) = sigma(k + 720) = sigma(k) + 720.

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A063680 Solutions to sigma(k) + 7 = sigma(k+7).

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