A049488 -id:A049488 - cp's OEIS frontend

A049492 Primes p such that p+4 and p+16 are also primes.

3, 7, 13, 37, 43, 67, 97, 163, 223, 277, 463, 487, 643, 757, 823, 937, 967, 1087, 1093, 1213, 1303, 1423, 1483, 1567, 1597, 1693, 1873, 2083, 2137, 2293, 2377, 2617, 2683, 2953, 3187, 3343, 3847, 3907, 4003, 4447, 4783, 5503, 5653, 5923, 6547, 6967, 6997

Offset: 1

Labos Elemer

nonn

All terms > 3 are == 1 (mod 6). - Zak Seidov, Sep 05 2014

Intersection of A023200 and A049488. - Michel Marcus, Sep 05 2014

			3, 3+4 = 7, 3+16 = 19 are all primes.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A023200, A049488, A049493.

Select[Prime[Range[900]],And@@PrimeQ[#+{4,16}]&] (* Harvey P. Dale, Jan 17 2011 *)

lista(nn) = forprime (n=1, nn, if (isprime(n+4) && isprime(n+16), print1(n, ", "))); \\ Michel Marcus, Sep 05 2014

A153418 Primes p such that p+18 is also prime.

Original entry on oeis.org

5, 11, 13, 19, 23, 29, 41, 43, 53, 61, 71, 79, 83, 89, 109, 113, 131, 139, 149, 163, 173, 179, 181, 193, 211, 223, 233, 239, 251, 263, 293, 313, 331, 349, 379, 383, 401, 421, 431, 439, 443, 449, 461, 491, 503, 523, 569, 599, 601, 613, 641, 643, 659, 673, 683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

Both p and p+18 have the same digital root (A010888). - Zak Seidov, Sep 14 2015
No term belongs to A030432. - Michel Marcus, Sep 14 2015
No term belongs to A045437. - Bruno Berselli, Sep 14 2015

			5 is in sequence because 5+18=23 is also prime;
11 is in sequence because 11+18=29 is also prime.

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

Cf. A007953, A010888, A045437, A049488, A030432, A153417.

A031936 is a subsequence. - Zak Seidov, Sep 13 2015

[p: p in PrimesUpTo(1000) | IsPrime(p+18)]; // Vincenzo Librandi, Apr 14 2013

lst={};d=18;Do[p=Prime[n];If[PrimeQ[p+d],AppendTo[lst,p]],{n,6!}];lst
Select[Prime[Range[150]], PrimeQ[(# + 18)]&] (* Vincenzo Librandi, Apr 14 2013 *)

list(n)=forprime(p=1,n,if(isprime(p+18),print1(p", ")))  \\ Anders Hellström, Sep 13 2015

[p for p in primes(700) if is_prime(p+18)]; # Bruno Berselli, Sep 14 2015

Definition improved by Bruno Berselli, Oct 31 2012

A031934 Lower prime of a pair of consecutive primes having a difference of 16.

Original entry on oeis.org

1831, 1933, 2113, 2221, 2251, 2593, 2803, 3121, 3373, 3391, 3433, 3643, 4057, 4111, 4567, 4621, 5023, 5281, 5623, 5881, 6637, 6763, 6841, 6883, 7333, 7417, 7993, 8581, 8647, 9013, 9241, 9661, 9907, 10273, 10513, 10867, 10957, 11197

Offset: 1

Jeff Burch

nonn

Conjecture: The sequence is infinite and for every n, a(n+1) < a(n)^(1+1/n); i.e., a(n)^(1/n) is a strictly decreasing function of n (see comment lines of the sequence A248855). - Jahangeer Kholdi and Farideh Firoozbakht, Nov 29 2014
All terms are == 1 mod 6. - Zak Seidov, Mar 27 2015
n such that A000720(n) = A000720(n-1)+1 = A000720(n+15) = A000720(n+16)-1. - Robert Israel, Mar 27 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A000720, A049488, A248855.

[p: p in PrimesUpTo(12000) | NextPrime(p)-p eq 16]; // Bruno Berselli, Apr 09 2013

P:= select(isprime,[seq(2*i+1,i=1..10000)]):
P[select(t -> P[t+1]-P[t]=16, [$1..nops(P)-1])]; # Robert Israel, Mar 27 2015

Transpose[Select[Partition[Prime[Range[1500]], 2, 1], Last[#] - First[#] == 16 &]][[1]] (* Bruno Berselli, Apr 09 2013 *)

is(n)=isprime(n) && nextprime(n+1)==n+16 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = prime(A320706(n)). - R. J. Mathar, Apr 30 2024

A049489 Primes p such that p + 32 is also prime.

Original entry on oeis.org

5, 11, 29, 41, 47, 71, 107, 131, 149, 167, 179, 191, 197, 239, 251, 281, 317, 347, 389, 401, 431, 467, 491, 509, 569, 587, 599, 641, 659, 677, 701, 719, 797, 821, 827, 887, 977, 1019, 1031, 1061, 1091, 1097, 1181, 1217, 1259, 1289, 1367, 1427, 1439, 1451

Offset: 1

Labos Elemer

			29 and 29 + 32 = 61 are both prime.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000

Cf. A001359, A023200, A023202, A049488.

[p: p in PrimesUpTo(2000) | IsPrime(p+32)]; // Vincenzo Librandi, Apr 22 2015

Primes:= select(isprime,{seq(i,i=3..10000,2)}):
sort(convert(Primes intersect map(`-`,Primes,32),list)); # Robert Israel, Dec 20 2015

Select[Range[2000], PrimeQ[#] && PrimeQ[# + 32] &] (* Vincenzo Librandi, Apr 22 2015 *)
Select[Prime[Range[300]],PrimeQ[#+32]&] (* Harvey P. Dale, Oct 14 2017 *)

isok(n) = isprime(n) && isprime(n+32); \\ Michel Marcus, Dec 31 2013

Name improved by Bruno Berselli, Apr 22 2015

A156104 Primes p such that p+36 is also prime.

Original entry on oeis.org

5, 7, 11, 17, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 101, 103, 113, 127, 131, 137, 157, 163, 191, 193, 197, 227, 233, 241, 257, 271, 277, 281, 311, 313, 317, 331, 337, 347, 353, 373, 383, 397, 421, 431, 443, 463, 467, 487, 521, 541, 557, 563, 571, 577, 607

Offset: 1

Vincenzo Librandi, Feb 08 2009

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

Cf. A156112.

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), A252089 (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), this sequence (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[p: p in PrimesUpTo(1000) | IsPrime(p + 36)]; // Vincenzo Librandi, Oct 31 2012

Select[Prime[Range[1000]], PrimeQ[(#+ 36)]&] (* Vincenzo Librandi, Oct 31 2012 *)

A153419 Primes p such that p+20 is also prime.

Original entry on oeis.org

3, 11, 17, 23, 41, 47, 53, 59, 83, 89, 107, 131, 137, 173, 179, 191, 251, 257, 263, 293, 311, 317, 347, 353, 359, 389, 401, 419, 443, 467, 479, 503, 521, 557, 587, 593, 599, 641, 653, 719, 809, 839, 857, 863, 887, 947, 971, 977, 1013, 1019, 1031, 1049, 1097

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

			3 is in the sequence because 3+20=23 is prime; 11 is in the sequence because 11+20=31 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A153417, A049488, A153418.

[p: p in PrimesUpTo(1100) | IsPrime(p + 20)]; // Vincenzo Librandi, Apr 14 2013

for a from 1 to 140 do if isprime(a) and isprime(a+20) then print(a)
  end if;  end do; # Matt C. Anderson, Jun 20 2022

Select[Prime[Range[200]], PrimeQ[(# + 20)]&] (* Vincenzo Librandi, Apr 14 2013 *)

Definition improved from Bruno Berselli, Oct 31 2012

A269257 Primes p such that p+2^4, p+2^6 and p+2^8 are all primes.

Original entry on oeis.org

7, 37, 163, 337, 757, 967, 1033, 1303, 2293, 2377, 2647, 2713, 3607, 5023, 6763, 7417, 8677, 8803, 9157, 9277, 10273, 14683, 14827, 15313, 15667, 16417, 20113, 21163, 21757, 22093, 24907, 27043, 27763, 29803, 29863, 32173, 34897, 36793, 36997, 37783, 38287, 38977, 39607

Offset: 1

Debapriyay Mukhopadhyay, Jul 12 2016

nonn

			The prime 7 is in the sequence because 7+16 = 23, 7+64 = 71 and 7+256 = 263 are all primes.
The prime 37 is in the sequence because 37+16 = 53, 37+64 = 101 and 37+256 = 293 are all primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Debapriyay Mukhopadhyay, C program to generate the terms of the sequences A269257, A269258, A269259, A269859 and A270203 up to 10^8

Subsequence of A002476, A049488, and A049490.

Select[Prime[Range[10000]], PrimeQ[# + 2^4] && PrimeQ[# + 2^6] && PrimeQ[# + 2^8]&] (* Jean-François Alcover, Jul 12 2016 *)
With[{c=2^Range[4,8,2]},Select[Prime[Range[4200]],AllTrue[#+c,PrimeQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 21 2017 *)

is(n)=n%6==1 && isprime(n+16) && isprime(n+64) && isprime(n+256) && isprime(n) \\ Charles R Greathouse IV, Jul 12 2016

use ntheory ":all"; say for sieve_prime_cluster(2,1e6, 16,64,256); # Dana Jacobsen, Jul 13 2016

A049488 INTERSECT A049490 INTERSECT A361483. - R. J. Mathar, Mar 26 2024

A252089 Primes p such that p + 26 is prime.

Original entry on oeis.org

3, 5, 11, 17, 41, 47, 53, 71, 83, 101, 113, 131, 137, 167, 173, 197, 251, 257, 281, 311, 347, 353, 383, 431, 461, 521, 587, 593, 617, 647, 683, 701, 743, 761, 797, 827, 857, 881, 911, 941, 971, 983, 1013, 1061, 1091, 1097, 1103, 1187, 1223, 1277, 1301, 1373

Offset: 1

Vincenzo Librandi, Dec 14 2014

			17 is in this sequence because 17+26 = 43 is prime.
431 is in this sequence because 431+26 = 457 is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), this sequence (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), A156104 (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+26)];

Select[Prime[Range[200]], PrimeQ[# + 26] &]

A361485 Primes p such that p + 1024 is also prime.

Original entry on oeis.org

7, 37, 67, 73, 79, 127, 139, 157, 163, 193, 199, 277, 283, 337, 349, 409, 457, 463, 487, 499, 547, 577, 613, 643, 673, 709, 787, 823, 853, 877, 883, 907, 1039, 1063, 1087, 1117, 1129, 1213, 1249, 1327, 1399, 1423, 1453, 1567, 1597, 1609, 1663, 1669, 1753, 1777, 1873, 1879

Offset: 1

Elmo R. Oliveira, Mar 13 2023

All terms are == 1 (mod 6).

			139 and 139 + 1024 = 1163 are both prime.

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A000040.

Cf. sequences of the type p + k are primes: A001359 (k = 2), A023200 (k = 4), A023202 (k = 8), A049488 (k = 16), A049489 (k = 32), A049490 (k = 64), A049491 (k = 128), A361483 (k = 256), A361484 (k = 512), this sequence (k = 1024).

lista(nn)=my(v=vector(nn), p=2); for(n=1, nn, until(isprime(p+1024), p=nextprime(p+1)); v[n]=p); v \\ Winston de Greef, Mar 20 2023

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

cp's OEIS Frontend

A049492 Primes p such that p+4 and p+16 are also primes.

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A153418 Primes p such that p+18 is also prime.

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Magma

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Sage

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A031934 Lower prime of a pair of consecutive primes having a difference of 16.

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Magma

Maple

Mathematica

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Formula

A049489 Primes p such that p + 32 is also prime.

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Magma

Maple

Mathematica

PARI

Extensions

A156104 Primes p such that p+36 is also prime.

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Magma

Mathematica

A153419 Primes p such that p+20 is also prime.

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Magma

Maple

Mathematica

Extensions

A269257 Primes p such that p+2^4, p+2^6 and p+2^8 are all primes.

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