A023202 -id:A023202 - cp's OEIS frontend

A046138 Primes p such that p+6 and p+8 are also primes.

5, 11, 23, 53, 101, 131, 173, 191, 233, 263, 563, 593, 653, 821, 1013, 1223, 1283, 1481, 1601, 1613, 1871, 2081, 2333, 2543, 2963, 3251, 3323, 3461, 3533, 3761, 3911, 3923, 4013, 4211, 4253, 4643, 4793, 5003, 5273, 5471, 5651, 5843, 5861, 6263, 6353, 6563

Offset: 1

Eric W. Weisstein

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Triplet.

Cf. A023201, A023202.

[p: p in PrimesUpTo(10^4)| IsPrime(p+6) and IsPrime(p+8)]; // Vincenzo Librandi, Jul 26 2015

for a from 3 by 2 to 10000 do
if `and`(isprime(a), isprime(a+6), isprime(a+8)) then print(a); end if;
end do; # Matt C. Anderson, Jul 24 2015

Select[Range@ 6000, AllTrue[{#, # + 6, # + 8}, PrimeQ] &] (* Michael De Vlieger, Jul 24 2015, Version 10 *)
Select[Prime[Range[1000]],AllTrue[#+{6,8},PrimeQ]&] (* Harvey P. Dale, Jun 05 2024 *)

use ntheory ":all"; say for sieve_prime_cluster(0,1e5, 6,8); # Dana Jacobsen, Oct 17 2017

A023201 INTERSECT A023202. - R. J. Mathar, Jan 23 2009

A087680 Numbers n such that n + 4 and n - 4 are both prime.

Original entry on oeis.org

7, 9, 15, 27, 33, 57, 63, 75, 93, 105, 135, 153, 177, 195, 237, 267, 273, 363, 393, 405, 435, 453, 483, 495, 567, 573, 597, 603, 657, 687, 705, 723, 747, 765, 825, 915, 933, 987, 1017, 1035, 1065, 1113, 1167, 1197, 1227, 1233, 1287, 1293, 1323, 1377, 1443

Offset: 1

Zak Seidov, Sep 27 2003

All terms > 7 (prime) are divisible by 3. Also note that n-4 and n+4 are not necessarily consecutive primes. First case when n-4 and n+4 are consecutive primes is for n=93 with n-4=89 and n+4=97. - Zak Seidov, Apr 22 2015

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A023202, A014574, A087678-A087683, A087695-A087697, A088764.

ZL:=[]:for p from 1 to 1444 do if (isprime(p) and isprime(p+8) ) then ZL:=[op(ZL),(p+(p+8))/2]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

f[n_]:=PrimeQ[n-4]&&PrimeQ[n+4]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,3,8!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)
Select[Prime[Range[250]],PrimeQ[#+8]&]+4 (* Harvey P. Dale, May 21 2023 *)

a(n) = A023202(n) + 4. - Michel Marcus, Apr 22 2015

More terms from Ray Chandler, Oct 26 2003

A104719 Concatenations of pairs of primes that differ by 10.

Original entry on oeis.org

313, 717, 1323, 1929, 3141, 3747, 4353, 6171, 7383, 7989, 97107, 103113, 127137, 139149, 157167, 163173, 181191, 223233, 229239, 241251, 271281, 283293, 307317, 337347, 349359, 373383, 379389, 409419, 421431, 433443, 439449, 457467, 499509

Offset: 1

Jonathan Vos Post and Parthasarathy Nambi, Mar 20 2005

There are no primes in this sequence after a(1) = 313, as all values thereafter are divisible by 3. Semiprimes in this sequence include: a(2) = 717 = 3 * 239, a(4) = 1929 = 3 * 643, a(6) = 3747 = 3 * 1249, a(7) = 4353 = 3 * 1451, a(10) = 7989 = 3 * 2663, a(11) = 97107 = 3 * 32369, a(13) = 127137 = 3 * 42379, a(17) = 181191 = 3 * 60397, a(18) = 223233 = 3 * 74411, a(29) = 421431 = 3 * 140477, a(30) = 433443 = 3 * 144481, a(34) = 547557 = 3 * 182519, a(35) = 577587 = 3 * 192529, a(40) = 691701 = 3 * 230567, a(41) = 709719 = 3 * 236573, a(49) = 919929 = 3 * 306643, a(52) = 10091019 = 3 * 3363673.

			Primes 3 and 13 differ by 10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A001358, A023202, A103195, A103206, A104718.

FromDigits[Join[IntegerDigits[#],IntegerDigits[#+10]]]&/@Select[ Prime[ Range[ 100]], PrimeQ[ #+10]&] (* Harvey P. Dale, Jun 14 2015 *)

a(n) = A023203(n) concatenated with A023203(n)+10.

A178099 Numbers k such that exactly three d in the range d <= k/2 exist which divide binomial(k-d-1,d-1) and which are not coprime to k.

Original entry on oeis.org

32, 38, 45, 51, 52, 56, 57, 63, 69, 87, 145, 209, 713, 1073, 3233, 3953, 5609, 8633, 11009, 18209, 23393, 31313, 38009, 56153, 71273, 74513, 131753, 154433, 164009, 189209, 205193, 233273, 245009, 321473, 328313, 356393, 363593, 431633, 471953, 497009

Offset: 1

Vladimir Shevelev, May 20 2010

nonn

Theorem: A number m > 145 is a member if and only if it is a product p*(p+8) such that both p and p+8 are primes (A023202).

The proof is similar to that of Theorem 1 in the Shevelev link. - Vladimir Shevelev, Feb 23 2016

Robert Price, Table of n, a(n) for n = 1..187
R. J. Mathar, Corrigendum to "On the divisibility...", arxiv:1109.0922 [math.NT], 2011.
Vladimir Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory, 3, no.1 (2007), 119-139.

Cf. A023202, A138389, A178071, A178098, A178101.

A178099 := proc(n) local dvs,d ; dvs := {} ; for d from 1 to n/2 do if gcd(n,d) > 1 and d in numtheory[divisors]( binomial(n-d-1,d-1)) then dvs := dvs union {d} ; end if; end do: if nops(dvs) = 3 then printf("%d,\n",n); end if; end proc:
for n from 1 do A178099(n) end do; # R. J. Mathar, May 28 2010

Select[Range[4000], Function[n, Count[Range[n/2], k_ /; And[! CoprimeQ[n, k], Divisible[Binomial[n - k - 1, k - 1], k]]] == 3]] (* Michael De Vlieger, Feb 17 2016 *)

isok(n) = sum(d=2, n\2, (gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0)) == 3; \\ Michel Marcus, Feb 17 2016

isok(n) = {my(nb = 0); for (d=2, n\2, if ((gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0), nb++); if (nb > 3, return (0));); nb == 3;} \\ Michel Marcus, Feb 17 2016

{k: A178101(k) = 3}.

Definition corrected, 54 and 91 removed by R. J. Mathar, May 28 2010
a(11)-a(23) from Michel Marcus, Feb 17 2016
a(24)-a(40) from Shevelev Theorem in Comments by Robert Price, May 14 2019

A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

Original entry on oeis.org

1, 1, 1, 1, 3556, 1, 34, 3, 4, 1, 2, 1, 11285, 5, 2, 124, 569, 1, 290, 3, 1, 165, 2, 1, 1, 2, 1, 316, 1, 2, 58957, 1, 3, 58617, 522, 2, 1, 1, 4, 1, 2, 1, 1, 2, 1, 7932, 4, 1, 5875, 1679, 4, 4, 3, 3, 1, 2, 307, 1, 1, 1, 1, 1, 4, 3206, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 2, 11170, 1, 2, 4245, 1, 1, 81, 2, 1, 1, 2, 58, 1, 3, 4, 7303, 1, 1, 5, 1, 3, 3, 3, 383, 111408, 1

Offset: 0

Farideh Firoozbakht and Jahangeer Kholdi, Nov 25 2014

nonn

All terms conjecturally are found. Note that according to the definition a(k,0) is the k-th term of the sequence {p | p is prime} namely for every positive integer k, a(k,0) = prime(k). Hence if Firoozbakht's conjecture is true then a(0)=1.

			a(0)=a(1)=a(2)=a(3)=1 conjecturally states that the four sequences A000040, A001359, A023200 and A023201 have this property: For every positive integer n, b(n) exists and b(n+1) < b(n)^(1+1/n). Namely b(n)^(1/n) is a strictly decreasing function of n.
If in the definition instead of the sequence {p | p and p+2n are primes} we set {p | p is prime and nextprime(p)=p+2n} then it seems that except for n=3 all terms of the new sequence {c(n)} are equal to 1 and for n=3, c(3)=7746. Note that c(3)=7746 means that the sequence {p | p is prime and nextprime(p)=p+6} = A031924 has this property: For all k >= 7746, A031924(k+1)^(1/(k+1)) < A031924(k)^(1/k).

Wikipedia, Firoozbakht's conjecture

Cf. A000040, A001359, A023200, A023201, A023202, A023203, A031924.

A049491 Numbers k such that k and k+128 are both prime.

Original entry on oeis.org

3, 11, 23, 29, 53, 71, 83, 101, 113, 149, 179, 239, 251, 269, 281, 293, 311, 359, 419, 443, 449, 479, 491, 503, 563, 599, 641, 659, 683, 701, 809, 839, 863, 881, 911, 941, 1103, 1109, 1151, 1163, 1193, 1301, 1319, 1361, 1439, 1451, 1481, 1493, 1499, 1571

Offset: 1

Labos Elemer

			11 and 11+128 = 139 are both prime.

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

Cf. A001359, A023200, A023202.

[n: n in [0..2000] | IsPrime(n) and IsPrime(n+128)]; // Vincenzo Librandi, Feb 02 2014

Select[Prime[Range[300]],PrimeQ[#+128]&] (* Harvey P. Dale, Jan 16 2011 *)

A104718 Concatenations of pairs of primes that differ by 8.

Original entry on oeis.org

311, 513, 1119, 2331, 2937, 5361, 5967, 7179, 8997, 101109, 131139, 149157, 173181, 191199, 233241, 263271, 269277, 359367, 389397, 401409, 431439, 449457, 479487, 491499, 563571, 569577, 593601, 599607, 653661, 683691, 701709, 719727, 743751

Offset: 1

Jonathan Vos Post, Mar 20 2005

There are no primes in this sequence after a(1) = 311, as all values thereafter are divisible by 3. Semiprimes in this sequence include: a(3) = 1119 = 3 * 373 a(6) = 5361 = 3 * 1787, a(8) = 7179 = 3 * 2393, a(9) = 8997 = 3 * 2999, a(10) = 101109 = 3 * 33703, a(13) = 173181 = 3 * 57727, a(15) = 233241 = 3 * 77747, a(17) = 269277 = 3 * 89759, a(21) = 431439 = 3 * 143813, a(26) = 569577 = 3 * 189859, a(35) = 821829 = 3 * 273943.

Cf. A000040, A001358, A023202, A103195, A103206, A104719.

a(n) = A023202(n) concatenated with A023202(n)+8.

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

A049436 p, p+8 and either p+2 or p+6 or both are all primes.

Original entry on oeis.org

3, 5, 11, 23, 29, 53, 59, 71, 101, 131, 149, 173, 191, 233, 263, 269, 431, 563, 569, 593, 599, 653, 821, 1013, 1031, 1061, 1223, 1229, 1283, 1289, 1319, 1451, 1481, 1601, 1613, 1619, 1871, 2081, 2129, 2333, 2339, 2381, 2543, 2549, 2711, 2789, 2963, 3251

Offset: 1

Labos Elemer

nonn

			3 is here because 5, 7 and 11 are primes; 5 is here because 7, 11 and 13 are primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A031926, A023202, A049437, A049438, A046134, A046138, A007530.

Select[Prime[Range[500]],PrimeQ[#+8]&&AnyTrue[#+{2,6},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2017 *)

cp's OEIS Frontend

A046138 Primes p such that p+6 and p+8 are also primes.

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A087680 Numbers n such that n + 4 and n - 4 are both prime.

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A104719 Concatenations of pairs of primes that differ by 10.

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A178099 Numbers k such that exactly three d in the range d <= k/2 exist which divide binomial(k-d-1,d-1) and which are not coprime to k.

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A248855 a(n) is the smallest positive integer m such that if k >= m then a(k+1,n)^(1/(k+1)) <= a(k,n)^(1/k), where a(k,n) is the k-th term of the sequence {p | p and p+2n are primes}.

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A049491 Numbers k such that k and k+128 are both prime.

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A104718 Concatenations of pairs of primes that differ by 8.

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A252089 Primes p such that p + 26 is prime.

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