A015916 -id:A015916 - cp's OEIS frontend

A023203 Primes p such that p + 10 is also prime.

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, 1021, 1039, 1051

Offset: 1

David W. Wilson

A subset of A002476. It appears that this is also a subset of A007645. The first few terms of A007645 that are not in this sequence are {67, 109, 151, 193, 199, 211, 277, 313, 331, 367, 397, 463, 487, 523, 541, 571, 601, 613, ...}. - Alexander Adamchuk, Aug 15 2006

The entries are all in A007645, because they cannot be of the form p = 3*j + 2. If they were, p + 10 = 3*j + 12 would be divisible by 3 and not prime. - R. J. Mathar, Oct 30 2009

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
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

Different from A015916. Cf. A031928, A079033.

Cf. A002476, A007645, A092146.

[n: n in [0..1000] | IsPrime(n) and IsPrime(n+10)]; // Vincenzo Librandi, Nov 20 2010

for p from 1 to 10000 do if isprime(p) and isprime(p+10) then print(p) end if end do # Matt C. Anderson, Aug 26 2022

Select[Prime[Range[200]], PrimeQ[# + 10] &] (* Harvey P. Dale, Dec 14 2011 *)

is(n)=isprime(n)&&isprime(n+10) \\ Charles R Greathouse IV, Jul 01 2013

Revised by N. J. A. Sloane, Jan 29 2013

New name from Michel Marcus, Mar 04 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

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

A271575 Primes p such that p+10, p+100 and p+1000 are all prime.

Original entry on oeis.org

13, 31, 97, 163, 181, 283, 409, 499, 709, 787, 811, 877, 1087, 1399, 1423, 1609, 1777, 1801, 1879, 2347, 2677, 2719, 3457, 3517, 3919, 4273, 4483, 5701, 6043, 6121, 6211, 6481, 6691, 7573, 8941, 9733, 9739, 10069, 10093, 10159, 10243, 10789, 11161, 11251, 11689, 12799, 12907

Offset: 1

Emre APARI, Apr 10 2016

nonn

Number of terms < 10^k: 0, 3, 12, 37, 159, 789, 3960, 21708, 129910, ..., . - Robert G. Wilson v, Jun 20 2018

			p=13; p+10=23 (is prime), p+100=113 (is prime), p+1000=1013 (is prime).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A002385, A015916, A023203.

select(t -> isprime(t+1000) and isprime(t+100) and isprime(t+10) and isprime(t), [seq(i,i=7..20000, 6)]); # Robert Israel, Jun 20 2018

Select[Prime[Range[10000]], PrimeQ[# + 10] && PrimeQ[# + 100] && PrimeQ[# + 1000] &] (* Robert Price, Apr 10 2016 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+10) && isprime(p+100) && isprime(p+1000), print1(p, ", "))); \\ Michel Marcus, Apr 10 2016

A271549 Primes p such that p+10^2, p+10^3, p+10^5, p+10^7, p+10^11, p+10^13 and p+10^17 are all prime.

Original entry on oeis.org

1399, 2157763, 13034041, 38208649, 38502313, 41518651, 42745111, 48154147, 49435063, 53872447, 58981513, 75194563, 83037247, 86139409, 101533963, 106287019, 140778403, 144593431, 155554237, 166083133, 166650193, 189371671, 199865893, 201738379, 224472877, 240133753, 271331773

Offset: 1

Emre APARI, Apr 10 2016

nonn

The exponents of 10 are all prime (2,3,5,7,11,13,17).

			p = 1399:
p+10^2  = 1499 (is prime).
p+10^3  = 2399 (is prime).
p+10^5  = 101399 (is prime).
p+10^7  = 10001399 (is prime).
p+10^11 = 100000001399 (is prime).
p+10^13 = 10000000001399 (is prime).
p+10^17 = 100000000000001399 (is prime).

Cf. A000040, A002385, A015916, A023203, A271575.

Select[Prime[Range[10^9]], PrimeQ[# + 10^2] && PrimeQ[# + 10^3] && PrimeQ[# + 10^5] &&  PrimeQ[# + 10^7] && PrimeQ[# + 10^11] &&  PrimeQ[# + 10^13] && PrimeQ[# + 10^17] &] (* Robert Price, Apr 10 2016 *)

lista(nn) = forprime(p=2, nn, if (isprime(p+10^2) && isprime(p+10^3) && isprime(p+10^5) && isprime(p+10^7) && isprime(p+10^11) && isprime(p+10^13) && isprime(p+10^17), print1(p, ", "))); \\ Altug Alkan, Apr 10 2016

More terms from Altug Alkan, Apr 10 2016

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A023203 Primes p such that p + 10 is also prime.

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

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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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A271575 Primes p such that p+10, p+100 and p+1000 are all prime.

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Maple

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PARI

A271549 Primes p such that p+10^2, p+10^3, p+10^5, p+10^7, p+10^11, p+10^13 and p+10^17 are all prime.

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