A023201 -id:A023201 - cp's OEIS frontend

A236509 Primes p with p + 2, p + 6 and prime(p) + 6 all prime.

5, 11, 107, 227, 311, 347, 821, 857, 1091, 1607, 1997, 2657, 3527, 4931, 5231, 8087, 8231, 9431, 10331, 11171, 12917, 13691, 13877, 21377, 22271, 24917, 27737, 29567, 32057, 33347, 35591, 36467, 37307, 39227, 42017

Offset: 1

Zhi-Wei Sun, Jan 27 2014

nonn

According to the conjecture in A236508, this sequence should have infinitely many terms.

			a(1) = 5 since 5, 5 + 2 = 7, 5 + 6 = 11 and prime(5) + 6 = 17 are all prime, but 2 + 2 = 4 and 3 + 6 = 9 are both composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A022004, A023201, A046117, A236464, A236508.

p[n_]:=p[n]=PrimeQ[n+2]&&PrimeQ[n+6]&&PrimeQ[Prime[n]+6]
n=0;Do[If[p[Prime[m]],n=n+1;Print[n," ",Prime[m]]],{m,1,10^6}]

A254041 Number of decompositions of 2n into an unordered sum of two sexy primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 3, 4, 2, 2, 4, 2, 3, 5, 3, 3, 5, 2, 4, 6, 2, 4, 6, 2, 4, 6, 4, 3, 6, 4, 3, 7, 4, 3, 8, 3, 4, 7, 3, 4, 7, 4, 5, 7, 5, 5, 9, 5, 5, 12, 4, 4, 10, 3, 5, 7, 4, 5, 6, 5, 6, 8, 4, 5, 9, 2, 5, 8, 3, 5, 8, 4, 4, 9, 6, 4, 9

Offset: 1

Lei Zhou, Jan 23 2015

"Sexy primes" are listed in A136207.

It is conjectured that a(n) > 0 for n > 4.

			When n = 79, 2n = 158 = 7 + 151 = 19 + 139 = 31 + 127 = 61 + 97 = 79 + 79 has five "two prime decompositions". Among the involved prime numbers 7, 19, 31, 61, 79, 97, 127, 139, 151, prime 127 and 139 are not sexy primes. So only three decompositions, 158 = 7 + 151 = 61 + 97 = 79 + 79 satisfy the definition of this sequence. Thus a(79) = 3.

Lei Zhou, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Math, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]
Wikipedia, Sexy Primes
Lei Zhou, Plot of a(n) up to n=1000000.

Cf. A136207, A023201, A240712, A002375.

Table[e = 2 n; ct = 0; p = 2; While[p = NextPrime[p]; p <= n, q = e - p; If[PrimeQ[q], If[(((p > 6) && PrimeQ[p - 6]) || PrimeQ[p + 6]) && (((q > 6) && PrimeQ[q - 6]) || PrimeQ[q + 6]), ct++]]]; ct, {n, 87}]

A261528 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m)+2 is prime with prime(prime(m)+2) = prime(prime(m))+6}.

Original entry on oeis.org

2, 891, 81002, 814812, 86050, 5917, 65527, 109853, 2563344, 25379, 2640232, 266076, 775889, 67387, 68111, 37950, 353416, 347139, 56390, 11299, 89491, 545458, 910786, 353416, 1913477, 9025, 111569, 511796, 1456228, 37909, 1494675, 212092, 69352, 107769, 300657, 1155675, 391972, 1073031, 55074, 49892

Offset: 1

Zhi-Wei Sun, Aug 23 2015

nonn

Conjecture: Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2 is prime with prime(prime(k)+2) = prime(prime(k))+6}.
This implies that there are infinitely many twin prime pairs {p, p+2} with prime(p+2) - prime(p) = 6.
Note that if prime(n+2)-prime(n) = 6 then prime(n+1)-prime(n) = 2 or 4.

			a(1) = 2 since 2*1 = 2, and prime(2)+2 = 3+2 = 5 is prime with prime(5)-prime(3) = 11-5 = 6.
a(2) = 891 since prime(891)+2 = 6947 + 2 = 6949 is prime with prime(6949)-prime(6947) = 70123-70117 = 6, and prime(891*2)+2 = 15269 + 2 = 15271 is prime with prime(15271)-prime(15269) = 167119-167113 = 6.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..20
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A001359, A006512, A023201, A046117, A259487, A259540, A261533.

f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,40}]

A261533 Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

Original entry on oeis.org

3, 5, 59, 2789, 5231, 6947, 8087, 11717, 15269, 16229, 17207, 17909, 18059, 18131, 24917, 28751, 35279, 37307, 39227, 39239, 41201, 43787, 45821, 47741, 51869, 53087, 53609, 58439, 64577, 69857, 70919, 75707, 79631, 84869, 92381, 93479, 96179, 102197, 102929, 106187

Offset: 1

Zhi-Wei Sun, Aug 23 2015

nonn

The conjecture in A261528 implies that the current sequence has infinitely many terms.

Note that for each n > 2 the difference prime(n+2)-prime(n) is at least 6.

			a(1) = 3 since 3 and 3+2 = 5 are twin prime, and prime(5)-prime(3) = 11-5 = 6.
a(2) = 5 since 5 and 5+2 = 7 are twin prime, and prime(7)-prime(5) = 17-11 = 6.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A001359, A006512, A023201, A046117, A236458, A259488, A259539, A261528.

f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
n=0;Do[If[PQ[k],n=n+1;Print[n," ",f[k]]],{k,1,10119}]
Select[Partition[Prime[Range[11000]],2,1],#[[2]]-#[[1]]==2&&Prime[#[[1]]+ 2]- Prime[#[[1]]]==6&][[All,1]] (* Harvey P. Dale, Apr 26 2020 *)

isok(i)=p=prime(i);isprime(p+2)&&prime(p+2)-prime(p)==6;
first(m)=my(v=vector(m));i=1;for(j=1,m,while(!isok(i),i++);v[j]=prime(i);i++);v; \\ Anders Hellström, Aug 23 2015

A275681 Table read by rows: list of sexy prime triples (p, p+6, p+12) such that p+18 is composite.

Original entry on oeis.org

7, 13, 19, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 73, 79, 97, 103, 109, 101, 107, 113, 151, 157, 163, 167, 173, 179, 227, 233, 239, 257, 263, 269, 271, 277, 283, 347, 353, 359, 367, 373, 379, 557, 563, 569, 587, 593, 599, 607, 613, 619, 647, 653, 659, 727, 733, 739

Offset: 1

Arkadiusz Wesolowski, Aug 05 2016

			The table starts:
7, 13, 19;
17, 23, 29;
31, 37, 43;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Wikipedia, Sexy prime
Index entries for primes, gaps between

Cf. A023201 (sexy primes), A046118, A123082, A275682.

lst:=[]; for p in PrimesUpTo(727) do b:=p+6; if IsPrime(b) then c:=b+6; if IsPrime(c) and not IsPrime(c+6) then lst:=lst cat [p, b, c]; end if; end if; end for; lst;

N:= 10^4: # to get all entries <= N
Primes:= select(isprime,{seq(i,i=1..N+18,2)}):
S:= select(`<=`, Primes,N) intersect map(t -> t-6, Primes) intersect map(t -> t-12, Primes) minus map(t -> t-18, Primes):
map(t ->(t,t+6,t+12), sort(convert(S,list))); # Robert Israel, Aug 05 2016

Most[#]&/@Select[Table[n+{0,6,12,18},{n,Prime[Range[200]]}],PrimeQ[#] == {True,True,True,False}&]//Flatten (* Harvey P. Dale, Jan 19 2017 *)

a(3*n-2) = A046118(n).
a(3*n-1) = A046118(n)+6.
a(3*n) = A046118(n)+12.

A275682 Table read by rows: list of sexy prime quadruples (p, p+6, p+12, p+18) such that p+24 is composite.

Original entry on oeis.org

11, 17, 23, 29, 41, 47, 53, 59, 61, 67, 73, 79, 251, 257, 263, 269, 601, 607, 613, 619, 641, 647, 653, 659, 1091, 1097, 1103, 1109, 1481, 1487, 1493, 1499, 1601, 1607, 1613, 1619, 1741, 1747, 1753, 1759, 1861, 1867, 1873, 1879, 2371, 2377, 2383, 2389

Offset: 1

Arkadiusz Wesolowski, Aug 05 2016

(5, 11, 17, 23, 29) is a sexy prime 5-tuple and this is the only sexy prime 5-tuple.

Essentially same as A123083.

			The table starts:
11, 17, 23, 29;
41, 47, 53, 59;
61, 67, 73, 79;
...

Vincenzo Librandi, Table of n, a(n) for n = 1..2100
Wikipedia, Sexy prime
Index entries for primes, gaps between

Cf. A023201, A023271, A123083, A275681.

lst:=[]; for p in PrimesInInterval(7, 2371) do b:=p+6; if IsPrime(b) then c:=b+6; if IsPrime(c) then d:=c+6; if IsPrime(d) then lst:=lst cat [p, b, c, d]; end if; end if; end if; end for; lst;

Most[#]&/@Select[Table[n + {0, 6, 12, 18, 24}, {n, Prime[Range[200]]}], PrimeQ[#]=={True, True, True, True, False}&]//Flatten (* Vincenzo Librandi, Jun 09 2017 *)
#+{0,6,12,18}&/@Select[Prime[Range[400]],AllTrue[#+{6,12,18},PrimeQ] && CompositeQ[#+24]&]//Flatten (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 29 2019 *)

a(n) = A123083(n+4).

A302087 Numbers k such that k^2+1 and (k+6)^2+1 are both prime.

Original entry on oeis.org

4, 10, 14, 20, 84, 110, 120, 124, 150, 170, 204, 224, 230, 250, 264, 300, 400, 430, 464, 490, 570, 674, 680, 690, 930, 960, 1004, 1054, 1060, 1140, 1144, 1150, 1314, 1410, 1434, 1550, 1564, 1570, 1580, 1654, 1784, 1870, 1964, 1974, 2050, 2074, 2080, 2120, 2260, 2304, 2314

Offset: 1

Seiichi Manyama, Mar 31 2018

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A005574, A023201, A096012, A302021.

[n: n in [1..2500] | IsPrime(n^2+1) and IsPrime((n+6)^2+1)]; // Vincenzo Librandi, Apr 02 2018

select(k->isprime(k^2+1) and isprime((k+6)^2+1),[$1..3000]); # Muniru A Asiru, Apr 02 2018

Select[Range[3000], PrimeQ[#^2 + 1] && PrimeQ[(# + 6)^2 + 1]&] (* Vincenzo Librandi, Apr 02 2018 *)

isok(k) = isprime(k^2+1) && isprime((k+6)^2+1); \\ Altug Alkan, Apr 02 2018

from sympy import isprime
k, klist, A302087_list = 0, [isprime(i**2+1) for i in range(6)], []
while len(A302087_list) < 10000:
    i = isprime((k+6)**2+1)
    if klist[0] and i:
        A302087_list.append(k)
    k += 1
    klist = klist[1:] + [i] # Chai Wah Wu, Apr 01 2018

A094231 Lesser member p of sexy primes (p, p+6) such that (p+1, p+2, p+3, p+4, p+5) all have the same number of prime divisors (counted with multiplicity).

Original entry on oeis.org

601, 42181, 70201, 240953, 277493, 414361, 418793, 619813, 632147, 637073, 723161, 732233, 739433, 761393, 781961, 879001, 934481, 979201, 1154233, 1320721, 1327673, 1357673, 1611361, 1685521, 1866233, 1877833, 1950457

Offset: 1

Jason Earls, May 29 2004

			42181 is a term because 42181 and 42187 are sexy primes while 42182-42186 each have 4 prime divisors (counting multiplicity).

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A023201, A046117.

f:=func; [p:p in PrimesUpTo(2000000)| IsPrime(p+6) and forall{k:k in [2..5]|f(p+k) eq f(p+1)} ]; // Marius A. Burtea, Dec 16 2019

Select[Range[2*10^6],AllTrue[{#,#+6},PrimeQ]&&Length[Union[ PrimeOmega[ Range[ #+1,#+5]]]]==1&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 16 2015 *)

A104873 Concatenations of pairs of primes that differ by 10^12.

Original entry on oeis.org

611000000000061, 1631000000000163, 1931000000000193, 2111000000000211, 2711000000000271, 3311000000000331, 5471000000000547, 6611000000000661, 7511000000000751, 7871000000000787, 9971000000000997, 10511000000001051

Offset: 1

Jonathan Vos Post, Mar 29 2005

Integers in this sequence can never be prime, as they are all multiples of 3. They can be semiprimes, as is the case for Prime(177) concatenated with Prime(37607912056) = 10511000000001051 = 3 * 3503666666667017.

			61 is prime, specifically prime(18) and 61 + 10^12 is prime, specifically prime(7607912020), so their concatenation is in this sequence: 611000000000061. The concatenation is not itself prime, as it equals 3 * 7 * 23 * 1265010351967.

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

Cf. A000040, A001358, A023201, A100750, A103195, A103206, A104718, A104719, A103523, A103534, A103576, A103617.

#*10^13+10^12+#&/@Select[Prime[Range[200]],PrimeQ[#+10^12]&] (* Harvey P. Dale, Jan 18 2021 *)

a(n) = Concatenate(P, P+10^12) iff P prime and P+10^12 prime.

A106059 Primes p such that p + 6 and 6*p + 1 are primes.

Original entry on oeis.org

5, 7, 11, 13, 17, 23, 37, 47, 61, 73, 83, 101, 103, 107, 131, 151, 173, 233, 257, 263, 271, 277, 311, 331, 347, 367, 373, 443, 461, 503, 557, 593, 601, 607, 641, 653, 727, 751, 853, 941, 947, 971, 1013, 1033, 1063, 1091, 1103, 1117, 1283, 1321, 1361

Offset: 1

Zak Seidov, May 07 2005

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007693, A023201.

[p: p in PrimesUpTo(5000)|IsPrime(p+6) and IsPrime(6*p+1)]; // Vincenzo Librandi, Jan 30 2011

Select[Prime[Range[220]], PrimeQ[6#+1]&&PrimeQ[1#+6]&]

cp's OEIS Frontend

A236509 Primes p with p + 2, p + 6 and prime(p) + 6 all prime.

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A254041 Number of decompositions of 2n into an unordered sum of two sexy primes.

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A261528 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m)+2 is prime with prime(prime(m)+2) = prime(prime(m))+6}.

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A261533 Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

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A275681 Table read by rows: list of sexy prime triples (p, p+6, p+12) such that p+18 is composite.

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Magma

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A275682 Table read by rows: list of sexy prime quadruples (p, p+6, p+12, p+18) such that p+24 is composite.

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Magma

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A302087 Numbers k such that k^2+1 and (k+6)^2+1 are both prime.

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