A023201 -id:A023201 - cp's OEIS frontend

A160370 Smaller member p of a pair (p,p+6) of consecutive primes in different centuries.

1097, 2897, 3797, 4597, 5297, 5897, 9397, 11497, 11897, 12197, 12497, 12697, 15797, 16097, 18797, 19597, 21997, 24097, 24197, 28597, 28697, 29297, 30097, 30197, 30697, 32497, 35597, 36997, 39097, 40897, 41597, 41897, 42397, 45497, 47297

Offset: 1

Ki Punches, May 11 2009

nonn

Note that the smaller member of a pair of sexy primes with the same constraint on centuries defines a different sequence, since members of a sexy prime pair do not need to be *consecutive* primes.
The larger member in the pair is obtained by adding 6 to an entry.
Every a(n)+3 is a multiple of 100 such that neither a(n)+2 nor a(n)+4 are primes. It appears that every integer occurs as the difference round((a(n+1)-a(n))/100); all numbers 1..333 occur as these differences for a(n) < 1000000000. - Hartmut F. W. Hoft, May 18 2017

			30097 + 6 = 30103.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A158277, A046117, A023201.

Transpose[Select[Partition[Prime[Range[5000]],2,1],#[[2]]-#[[1]] == 6 && Floor[#[[1]]/100]!=Floor[#[[2]]/100]&]][[1]] (* Harvey P. Dale, Apr 28 2012 *)
a160370[n_] := Select[Range[97, n, 100], AllTrue[# + {0, 6}, PrimeQ] && NoneTrue[# + {2, 4}, PrimeQ]&]
a160370[49000] (* data *) (* Hartmut F. W. Hoft, May 18 2017 *)

{A031924(n): [A031924(n)/100] <> [A031925(n)/100]} where [..]=floor(..).

Edited by R. J. Mathar, May 14 2009

A178098 Numbers n such that exactly two positive d in the range d <= n/2 exist which divide binomial(n-d-1, d-1) and which are not coprime to n.

Original entry on oeis.org

26, 30, 36, 40, 42, 44, 91, 95, 115, 119, 133, 161, 187, 247, 391, 667, 1147, 1591, 1927, 2491, 3127, 4087, 4891, 5767, 7387, 9991, 10807, 11227, 12091, 17947, 23707, 25591, 28891, 30967, 37627, 38407, 51067, 52891, 55687, 64507, 67591, 70747, 75067, 78391

Offset: 1

Vladimir Shevelev, May 20 2010

nonn

Theorem: A number m > 161 is a member if and only if it is a product p*(p+6) such that both p and p+6 are primes (A023201). 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..353
R. J. Mathar, Corrigendum to "On the divisibility of...", arXiv:1109.0922 [math.NT], 2011.
V. Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory 3, no.1 (2007), 119-139.

Cf. A178101, A178071, A138389, A023201, A178099.

Select[Range@ 4000, Function[n, Count[Range[n/2], k_ /; And[! CoprimeQ[n, k], Divisible[Binomial[n - k - 1, k - 1], k]]] == 2]] (* Michael De Vlieger, 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 > 2, return (0));); nb == 2; \\ Michel Marcus, Feb 17 2016

{n: A178101(n) = 2}.

91 inserted by R. J. Mathar, May 28 2010
a(18)-a(36) from Michel Marcus, Feb 17 2016
a(37)-a(44) (based on theorem from Vladimir Shevelev in Comments) from Robert Price, May 14 2019

A098428 Number of sexy prime pairs (p, p+6) with p <= n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Since there are 2 congruence classes of sexy prime pairs, (-1, -1) (mod 6) and (+1, +1) (mod 6), the number of sexy prime pairs up to n is the sum of the number of sexy prime pairs for each class, expected to be asymptotically the same for both (with the expected Chebyshev bias against the quadratic residue class (+1, +1) (mod 6), which doesn't affect the asymptotic distribution among the 2 classes). - Daniel Forgues, Aug 05 2009

			The first sexy prime pairs are: (5,11), (7,13), (11,17), (13,19), ...
therefore the sequence starts: 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, ...

Daniel Forgues, Table of n, a(n) for n=1..99994
Eric Weisstein's World of Mathematics, 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]

Cf. A023201, A046117, A098424, A071538, A098429.

Accumulate[Table[If[PrimeQ[n]&&PrimeQ[n+6],1,0],{n,100}]] (* Harvey P. Dale, Feb 08 2015 *)

apply( {A098428(n,o=2,q=o,c)=forprime(p=1+q, n+6, (o+6==p)+((o=q)+6==q=p) && c++);c}, [1..99]) \\ M. F. Hasler, Jan 02 2020
[#[p:p in PrimesInInterval(1,n)| IsPrime(p+6)]:n in [1..100]]; // Marius A. Burtea, Jan 03 2020

a(n) = # { p in A023201 | p <= n } = number of elements in intersection of A023201 and [1,n]. - M. F. Hasler, Jan 02 2020

Edited by Daniel Forgues, Aug 01 2009, M. F. Hasler, Jan 02 2020

A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6k-1 and 6k+1 twin primes.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 2, 12, 39, 98, 526, 2117, 6663, 15043, 68403, 791581, 4826577, 19592777, 102551299, 739788968, 4449585790, 36547266589, 324446266072, 2743681178070

Offset: 1

Zhi-Wei Sun, Sep 08 2013

Conjecture: a(n) > 0 for all n > 0.
This implies the twin prime conjecture, and it is similar to the prime circle problem mentioned in A051252.
For each n = 2,3,... construct an undirected simple graph T(n) with vertices 0,1,...,n which has an edge connecting two distinct vertices i and j if and only if 6*(i+j)-1 and 6*(i+j)+1 are twin primes. Then a(n) is just the number of Hamiltonian cycles contained in T(n). Thus a(n) > 0 if and only if T(n) is a Hamilton graph.
Zhi-Wei Sun also made the following similar conjectures for odd primes, Sophie Germain primes, cousin primes and sexy primes:
(1) For any integer n > 0, there is a permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are integers of the form (p-1)/2, where p is an odd prime. Also, we may replace the above (p-1)/2 by (p+1)/4 or (p-1)/6; when n > 4 we may substitute (p-1)/4 for (p-1)/2.
(2) For any integer n > 2, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are integers of the form (p+1)/6, where p is a Sophie Germain prime.
(3) For any integer n > 3, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are among those integers k with 6*k+1 and 6*k+5 both prime.
(4) For any integer n > 4, there is a permutation i_0, i_1, ..., i_n of 0, 1,..., n  such that i_0+i_1, i_1+i_2, ..., i_{n-1}+i_n, i_n+i_0 are among those integers k with 2*k-3 and 2*k+3 both prime.

			a(n) = 1 for n = 1,2,3 due to the permutation (0,...,n).
a(4) = 2 due to the permutations (0,1,4,3,2) and (0,2,1,4,3).
a(5) = 2 due to the permutations (0,1,4,3,2,5), (0,3,4,1,2,5).
a(6) = 2 due to the permutations
  (0,1,6,4,3,2,5) and (0,3,4,6,1,2,5).
a(7) = 5 due to the permutations
  (0,1,6,4,3,2,5,7), (0,1,6,4,3,7,5,2), (0,2,1,6,4,3,7,5),
  (0,3,4,6,1,2,5,7), (0,5,2,1,6,4,3,7).
a(8) = 2 due to the permutations
  (0,1,6,4,8,2,3,7,5) and (0,1,6,4,8,2,5,7,3).
a(9) = 12 due to the permutations
  (0,1,6,4,3,9,8,2,5,7), (0,1,6,4,8,9,3,2,5,7),
  (0,1,6,4,8,9,3,7,5,2), (0,2,1,6,4,8,9,3,7,5),
  (0,2,8,9,1,6,4,3,7,5), (0,3,4,6,1,9,8,2,5,7),
  (0,3,9,1,6,4,8,2,5,7), (0,3,9,8,4,6,1,2,5,7),
  (0,5,2,1,6,4,8,9,3,7), (0,5,2,8,4,6,1,9,3,7),
  (0,5,2,8,9,1,6,4,3,7), (0,5,7,3,9,1,6,4,8,2).
a(10) > 0 due to the permutation (0,5,2,3,9,1,6,4,8,10,7).
a(11) > 0 due to the permutation (0,10,8,9,3,7,11,6,4,1,2,5).
a(12) > 0 due to the permutation
        (0, 5, 2, 1, 6, 4, 3, 9, 8, 10, 7, 11, 12).

Zhi-Wei Sun, Twin primes and circular permutations, a message to Number Theory List, Sept. 8, 2013.
Z.-W. Sun, Some new problems in additive combinatorics, arXiv preprint arXiv:1309.1679 [math.NT], 2013-2014.

Cf. A000040, A001359, A006512, A023200, A046132, A023201, A046117, A005384, A051252, A228766, A228860, A228886.

(* A program to compute required circular permutations for n = 7. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (0,7,5,2,3,4,6,1) is identical to (0,1,6,4,3,2,5,7) if we ignore direction. Thus a(7) is half of the number of circular permutations yielded by this program. *)
tp[n_]:=tp[n]=PrimeQ[6n-1]&&PrimeQ[6n+1]
V[i_]:=Part[Permutations[{1,2,3,4,5,6,7}],i]
m=0
Do[Do[If[tp[If[j==0,0,Part[V[i],j]]+If[j<7,Part[V[i],j+1],0]]==False,Goto[aa]],{j,0,7}];
m=m+1;Print[m,":"," ",0," ",Part[V[i],1]," ",Part[V[i],2]," ",Part[V[i],3]," ",Part[V[i],4]," ",Part[V[i],5]," ",Part[V[i],6]," ",Part[V[i],7]];Label[aa];Continue,{i,1,7!}]

a(10)-a(25) from Max Alekseyev, Sep 12 2013

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.

A046119 Middle member of a sexy prime triple: value of p+6 such that p, p+6 and p+12 are all prime, but p+18 is not (although p-6 might be).

Original entry on oeis.org

13, 23, 37, 53, 73, 103, 107, 157, 173, 233, 263, 277, 353, 373, 563, 593, 613, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1283, 1297, 1367, 1433, 1453, 1493, 1613, 1663, 1753, 1783, 1873, 1907, 1993, 2137, 2287, 2383, 2417, 2683, 2693, 2713

Offset: 1

Eric W. Weisstein

nonn

p-6 will be prime if the prime triple contains the last 3 primes of a sexy prime quadruple.

If a sexy prime triple happens to include the last 3 members of a sexy prime quadruple, this sequence will contain the sexy prime triple's middle member; e.g., a(4)=53 is the middle member of the sexy prime triple (47, 53, 59), but is also the third member of the sexy prime quadruple (41, 47, 53, 59). - Daniel Forgues, Aug 05 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, 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].

Cf. A023201, A046117.

Cf. A046118, A046120.

lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&!PrimeQ[p+18], AppendTo[lst, p+6]], {n, 7!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[400]],And@@PrimeQ[{#-6,#+6}]&&!PrimeQ[#+12]&] (* Harvey P. Dale, Nov 01 2011 *)

a(n) = A046118(n) + 6. - Michel Marcus, Jan 06 2015

Definition edited by Daniel Forgues, Aug 12 2009

A046120 Largest member of a sexy prime triple; value of p+12 where p, p+6 and p+12 are all prime, but p+18 is not.

Original entry on oeis.org

19, 29, 43, 59, 79, 109, 113, 163, 179, 239, 269, 283, 359, 379, 569, 599, 619, 659, 739, 953, 983, 1109, 1129, 1193, 1229, 1289, 1303, 1373, 1439, 1459, 1499, 1619, 1669, 1759, 1789, 1879, 1913, 1999, 2143, 2293, 2389, 2423, 2689, 2699, 2719

Offset: 1

Eric W. Weisstein

nonn

If a sexy prime triple happens to include the last 3 members of a sexy prime quadruple, this sequence will contain the sexy prime triple's largest member; e.g., a(4)=59 is the largest member of the sexy prime triple (47, 53, 59), but is the fourth member of the sexy prime quadruple (41, 47, 53, 59). - Daniel Forgues, Aug 05 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, 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].

Cf. A023201, A046117.

Cf. A046118, A046119.

lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&!PrimeQ[p+18], AppendTo[lst, p+12]], {n, 7!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
#+12&/@Select[Prime[Range[400]],PrimeQ[#+{6,12,18}]=={True,True,False}&] (* Harvey P. Dale, Dec 08 2012 *)

a(n) = A046118(n)+12 and a(n) = A046119(n)+6. - Michel Marcus, Jan 06 2015

A046122 Second member of a sexy prime quadruple: value of p+6 such that p, p+6, p+12 and p+18 are all prime.

Original entry on oeis.org

11, 17, 47, 67, 257, 607, 647, 1097, 1487, 1607, 1747, 1867, 2377, 2677, 3307, 3917, 4007, 5107, 5387, 5437, 5647, 6317, 6367, 9467, 11827, 12107, 12647, 13457, 14627, 14747, 15797, 15907, 17477, 18217, 19477, 20347, 21487, 23327, 24097

Offset: 1

Eric W. Weisstein

nonn

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

Cf. A023201, A023271, A046117, A046123, A046124.

lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&PrimeQ[p+18], AppendTo[lst, p+6]], {n, 8!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)

a(n) = 6 + A023271(n) = A046123(n) - 6. - R. J. Mathar, Jun 28 2012

A046123 Third member of a sexy prime quadruple: value of p+12 such that p, p+6, p+12 and p+18 are all prime.

Original entry on oeis.org

17, 23, 53, 73, 263, 613, 653, 1103, 1493, 1613, 1753, 1873, 2383, 2683, 3313, 3923, 4013, 5113, 5393, 5443, 5653, 6323, 6373, 9473, 11833, 12113, 12653, 13463, 14633, 14753, 15803, 15913, 17483, 18223, 19483, 20353, 21493, 23333, 24103

Offset: 1

Eric W. Weisstein

nonn

Is 17 the only term that is not equal to 3 mod 10? It is the only such term up to the one millionth prime. - Harvey P. Dale, Jan 25 2023

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

Cf. A023201, A046117.

Cf. A023271, A046122, A046124.

lst={};Do[p=Prime[n];If[PrimeQ[p+6]&&PrimeQ[p+12]&&PrimeQ[p+18], AppendTo[lst, p+12]], {n, 8!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
Select[Prime[Range[3000]],AllTrue[#+{-12,-6,6},PrimeQ]&] (* Harvey P. Dale, Jan 25 2023 *)

a(n) = A046122(n) + 6. - Amiram Eldar, Apr 22 2022

A049482 Primes p such that p + 210 is also prime.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 41, 47, 53, 59, 61, 67, 71, 73, 83, 97, 101, 103, 107, 127, 137, 139, 149, 157, 163, 173, 179, 191, 199, 211, 223, 229, 233, 239, 251, 257, 269, 277, 281, 293, 311, 313, 331, 337, 347, 353, 359, 367, 383, 389, 397, 409, 421, 431, 433

Offset: 1

Labos Elemer

nonn

			Both 13 and 13 + 210 = 223 are prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A045320, A001359, A023201, A049481.

Select[Prime@ Range@ 84, PrimeQ[# + 210] &] (* Michael De Vlieger, Jun 29 2017 *)

list(lim)=my(v=List()); forprime(p=2,lim, if(isprime(p+210), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 23 2017

a(n) >> n log^2 n. - Charles R Greathouse IV, Feb 23 2017

cp's OEIS Frontend

A160370 Smaller member p of a pair (p,p+6) of consecutive primes in different centuries.

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A178098 Numbers n such that exactly two positive d in the range d <= n/2 exist which divide binomial(n-d-1, d-1) and which are not coprime to n.

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A098428 Number of sexy prime pairs (p, p+6) with p <= n.

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A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6*k-1 and 6*k+1 twin primes.

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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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A046119 Middle member of a sexy prime triple: value of p+6 such that p, p+6 and p+12 are all prime, but p+18 is not (although p-6 might be).

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A046120 Largest member of a sexy prime triple; value of p+12 where p, p+6 and p+12 are all prime, but p+18 is not.

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A046122 Second member of a sexy prime quadruple: value of p+6 such that p, p+6, p+12 and p+18 are all prime.

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A228917 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that i_0+i_1, i_1+i_2, ...,i_{n-1}+i_n, i_n+i_0 are among those k with 6k-1 and 6k+1 twin primes.