A023201 -id:A023201 - cp's OEIS frontend

A172295 Partial sums of A023201.

5, 12, 23, 36, 53, 76, 107, 144, 185, 232, 285, 346, 413, 486, 569, 666, 767, 870, 977, 1108, 1259, 1416, 1583, 1756, 1947, 2140, 2363, 2590, 2823, 3074, 3331, 3594, 3865, 4142, 4449, 4760, 5091, 5438, 5791, 6158, 6531, 6914, 7347, 7790, 8247

Offset: 1

Jonathan Vos Post, Jan 30 2010

A023201 is also known as the smaller numbers of pairs of sexy primes. The subsequence of prime partial sums of smaller numbers of pairs of sexy primes begins 5, 23, 53, 107, 569, 977, 1259, 1583, 3331. The subsubsequence of smaller numbers of pairs of sexy prime partial sums of smaller numbers of pairs of sexy primes begins 5, 107, 977. This is to smaller members of sexy prime pairs as A172112 is to A023200 smaller member p of cousin prime pairs (p, p+4)

			a(19) = 5 + 7 + 11 + 13 + 17 + 23 + 31 + 37 + 41 + 47 + 53 + 61 + 67 + 73 + 83 + 97 + 101 + 103 + 107 = 977, which is A023201(73).

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

Cf. A000040, A023201, A172112.

Accumulate[Select[Prime[Range[120]],PrimeQ[#+6]&]] (* Harvey P. Dale, Jan 18 2020 *)

More terms from Sean A. Irvine, Sep 27 2011

A338812 Smaller term of a pair of sexy primes (A023201) such that the distance to next pair (A227346) is a square.

Original entry on oeis.org

7, 13, 37, 97, 103, 223, 307, 331, 457, 541, 571, 853, 877, 1087, 1297, 1423, 1483, 1621, 1867, 1993, 2683, 3457, 3511, 3691, 3761, 3847, 4513, 4657, 4783, 4951, 5227, 5521, 5647, 5861, 6337, 6547, 6823, 7481, 7541, 7681, 7717, 7753, 7873, 8287, 8521, 8887, 9007, 9397, 10267, 10453

Offset: 1

Claude H. R. Dequatre, Nov 10 2020

Considering the 10^6 sexy prime pairs from (5,11) to (115539653,115539659), we note the following:
65340 sequence terms (6.5%) are linked to a distance between two consecutive sexy prime pairs which is a square.
List of the 16 classes of distances which are squares: 4,16,36,64,100,144,196,256,324,400,484,576,676,784,900,1024.
The frequency of the distances which are squares decreases when their size increases, with a noticeable higher frequency for the distance 36.
First 20 distances which are squares with in parentheses the subtraction of the smallest members of the related two consecutive sexy prime pairs: 4 (11-7), 4 (17-13),4 (41-37),4 (101-97),4 (107-103),4 (227-223),4 (311-307), 16 (347-331),4 (461-457),16 (557-541),16 (587-571),4 (857-853), 4 (881-877), 4 (1091-1087),4 (1301-1297),4 (1427-1423),4 (1487-1483),36 (1657-1621), 4 (1871-1867),4 (1997-1993).

			a(2)=13 is in the sequence because the two consecutive sexy prime pairs being (13,19) and (17,23),the distance between them is 17-13=4 which is a square (2^2).
73 is not in the sequence because the two consecutive sexy prime pairs being (73,79) and (83,89),the distance between them is 83-73=10 which is not a square.

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

Cf. A023201, A046117,A227346, A000290, A161002, A161533, A161534, A138198.

count:= 0: sp:= 5: R:= NULL:
p:= sp;
while count < 100  do
    p:= nextprime(p);
    if isprime(p+6) then
      d:= p - sp;
      if issqr(d) then
        count:= count+1; R:= R, sp;
      fi;
      sp:= p;
    fi;
od:
R; # Robert Israel, May 08 2024

lista(nn) = {my(vs = select(x->(isprime(x) && isprime(x+6)), [1..nn]), vd = vector(#vs-1, k, vs[k+1] - vs[k]), vk = select(issquare, vd, 1)); vector(#vk, k, vs[vk[k]]);} \\ Michel Marcus, Nov 14 2020

primes<-generate_n_primes(7000000)
Matrix_1<-matrix(c(primes),nrow=7000000,ncol=1,byrow=TRUE)
p1<-c(0)
p2<-c(0)
k<-c(0)
distance<-c(0)
distance_square<-(0)
Matrix_2<-cbind(Matrix_1,p1,p2,k,distance,distance_square)
counter=0
j=1
while(j<= 7000000){
  p<-(Matrix_2[j,1])+6
  if(is_prime(p)){
    counter=counter+1
    Matrix_2[counter,2]<-(p-6)
    Matrix_2[counter,3]<-p
  }
  j=j+1
}
a_n<-c()
k=1
while(k<=1000000){
  Matrix_2[k,4]<-k
  dist<-Matrix_2[k+1,2]-Matrix_2[k,2]
  Matrix_2[k,5]<-dist
  if(sqrt(dist)%%1==0){
    Matrix_2[k,6]<-dist
    a_n<-append(a_n,Matrix_2[k,2])
  }
  k=k+1
}
View(Matrix_2)
View(a_n)

A255472 Number of decompositions of 2n into sums of two primes p <= q such that one or both p and q are elements of A023201.

Original entry on oeis.org

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

Offset: 1

Lei Zhou, Feb 23 2015

Conjecture: for all n > 3, a(n) > 0.
If 2n = p + q and p+6 is also a prime, 2n+6 can be written as the sum of two primes p+6 and q.
The conjecture is weaker than a conjecture of Sun posed in 2012 (see A219055). - Zhi-Wei Sun, Mar 18 2015

			n=4: 2n=8=3+5, 5+6=11 is also a prime number. This is the only occurrence, so a(4)=1.
n=5: 2n=10=3+7=5+5. Both 5+6=11 and 7+6=13 are prime numbers. Two occurrences found, so a(5)=2.

Lei Zhou, Table of n, a(n) for n = 1..10000
Lei Zhou, Plot for 0
Index entries for sequences related to Goldbach conjecture

Cf. A023201, A045917, A199920, A219055.

Table[e = 2 n; ct = 0; p1 = 2; While[p1 = NextPrime[p1]; p1 <= n, p2 = e - p1; If[PrimeQ[p2], If[PrimeQ[p1 + 6] || PrimeQ[p2 + 6], ct++]]]; ct, {n, 1, 100}]

a(n) = {nb = 0; forprime(p=2, 2*n, if ((q=2*n-p) && (q <= p) && isprime(q=2*n-p) && (isprime(q+6) || isprime(p+6)), nb++);); nb;} \\ Michel Marcus, Mar 01 2015

A022004 Initial members of prime triples (p, p+2, p+6).

Original entry on oeis.org

5, 11, 17, 41, 101, 107, 191, 227, 311, 347, 461, 641, 821, 857, 881, 1091, 1277, 1301, 1427, 1481, 1487, 1607, 1871, 1997, 2081, 2237, 2267, 2657, 2687, 3251, 3461, 3527, 3671, 3917, 4001, 4127, 4517, 4637, 4787, 4931, 4967, 5231, 5477

Offset: 1

Warut Roonguthai

Subsequence of A001359. - R. J. Mathar, Feb 10 2013
All terms are congruent to 5 (mod 6). - Matt C. Anderson, May 22 2015
Intersection of A001359 and A023201. - Zak Seidov, Mar 12 2016

Matt C. Anderson Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
T. Forbes and Norman Luhn, Prime k-tuplets
R. J. Mathar, Table of Prime Gap Constellations (2013,2024), 275 pages (no not print...)
Thomas R. Nicely, Enumeration of the prime triples (q,q+2,q+6) to 1e16.
P. Pollack, Analytic and Combinatorial Number Theory, Course Notes, p. 132, ex. 3.4.3. [Broken link?]
P. Pollack, Analytic and Combinatorial Number Theory, Course Notes, p. 132, ex. 3.4.3.
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, Prime Triplet
Index entries for primes, gaps between

Cf. A073648, A098412, A372247 (subsequence).
Cf. A001359, A023201.
Subsequence of A007529.

[ p: p in PrimesUpTo(10000) | IsPrime(p+2) and IsPrime(p+6) ] // Vincenzo Librandi, Nov 19 2010

A022004 := proc(n)
    if n= 1 then
        5;
    else
        for a from procname(n-1)+2 by 2 do
            if isprime(a) and isprime(a+2) and isprime(a+6) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 11 2012

Select[Prime[Range[1000]], PrimeQ[#+2] && PrimeQ[#+6]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)
Transpose[Select[Partition[Prime[Range[1000]],3,1],Differences[#]=={2,4}&]][[1]] (* Harvey P. Dale, Dec 24 2011 *)

is(n)=isprime(n)&&isprime(n+2)&&isprime(n+6) \\ Charles R Greathouse IV, Jul 01 2013

from sympy import primerange
def aupto(limit):
  p, q, alst = 2, 3, []
  for r in primerange(5, limit+7):
    if p+2 == q and p+6 == r: alst.append(p)
    p, q = q, r
  return alst
print(aupto(5477)) # Michael S. Branicky, May 11 2021

A046117 Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.)

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 89, 103, 107, 109, 113, 137, 157, 163, 173, 179, 197, 199, 229, 233, 239, 257, 263, 269, 277, 283, 313, 317, 337, 353, 359, 373, 379, 389, 439, 449, 463, 467, 509, 547, 563, 569, 577, 593, 599, 607, 613

Offset: 1

Eric W. Weisstein

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
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].
Wikipedia, Sexy prime.

Cf. A023201, A098428, A087695.

[p:p in PrimesInInterval(7,650)| IsPrime(p-6)]; // Marius A. Burtea, Jan 03 2020

q=6; a={}; Do[p1=Prime[n]; p2=p1+q; If[PrimeQ[p2], AppendTo[a, p2]], {n, 7^2}]; a "and/or" Select[Prime[Range[3, 7^2]], PrimeQ[ # ]&&PrimeQ[ #-6]&] (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Select[Prime[Range[4,200]],PrimeQ[#-6]&] (* Harvey P. Dale, Mar 31 2018 *)

forprime(p=2,1e4,if(isprime(p-6),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A087695(n) + 3.

a(n) = A023201(n) + 6. - M. F. Hasler, Jan 02 2020

Name edited by M. F. Hasler, Jan 02 2020

A031924 Primes followed by a gap of 6, i.e., next prime is p + 6.

Original entry on oeis.org

23, 31, 47, 53, 61, 73, 83, 131, 151, 157, 167, 173, 233, 251, 257, 263, 271, 331, 353, 367, 373, 383, 433, 443, 503, 541, 557, 563, 571, 587, 593, 601, 607, 647, 653, 677, 727, 733, 751, 941, 947, 971, 977, 991, 1013, 1033, 1063, 1097, 1103, 1117, 1123, 1181

Offset: 1

Jeff Burch

nonn

Original name: Lower prime of a difference of 6 between consecutive primes.

Conjecture: The sequence is infinite and for every n >= 7746, a(n+1) < a(n)^(1+1/n). Namely for n >= 7746, 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

			23 is a term as the next prime 29 = 23 + 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020
Index entries for primes, gaps between

Cf. A001359, A023201, A031925; A031924 and A007529 together give A023201.

P:=Filtered([1..1200],IsPrime);;
List(Filtered([1..Length(P)-1],i->P[i+1]-P[i]=6),k->P[k]); # Muniru A Asiru, Jan 30 2019

[p: p in PrimesUpTo(1200) | NextPrime(p)-p eq 6]; // Bruno Berselli, Apr 09 2013

A031924 := proc(n)
    option remember;
    if n = 1 then
        return 23;
    else
        p := nextprime(procname(n-1)) ;
        q := nextprime(p) ;
        while q-p <> 6 do
            p := q ;
            q := nextprime(p) ;
        end do:
        return p;
    end if;
end proc: # R. J. Mathar, Jan 23 2013

Transpose[Select[Partition[Prime[Range[200]], 2, 1], Last[#] - First[#] == 6 &]][[1]] (* Bruno Berselli, Apr 09 2013 *)

is(n)=isprime(n)&&nextprime(n+1)-n==6 \\ Charles R Greathouse IV, Mar 21 2013

apply( A031924(n,p=2,show=0,g=6)={forprime(q=p+1,, p+g!=(p=q) || (show&&print1(p-g",")) || n-- || return(p-g))}, [1..99]) \\ Use nxt(p)=A031924(1,p) to get the term following p, use show=1 to print all a(1..n), g to select a different gap. - M. F. Hasler, Jan 02 2020

New name from M. F. Hasler, Jan 02 2020

A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 37, 41, 67, 97, 101, 103, 107, 191, 193, 223, 227, 277, 307, 311, 347, 457, 461, 613, 641, 821, 823, 853, 857, 877, 881, 1087, 1091, 1277, 1297, 1301, 1423, 1427, 1447, 1481, 1483, 1487, 1607, 1663, 1693, 1783, 1867, 1871, 1873, 1993, 1997

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Or, prime(m) such that prime(m+2) = prime(m)+6. - Zak Seidov, May 07 2012

H. Riesel, "Prime numbers and computer methods for factorization", Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Cf. A031924, A023201, A098414, A098415, A098424, A098416, A098417.

Union of A022004 and A022005.

[NthPrime(n): n in [1..310] | (NthPrime(n)+6) eq NthPrime(n+2)]; // Bruno Berselli, May 07 2012

N:= 10000: # to get all terms <= N
Primes:= select(isprime, [seq(2*i+1, i=1..floor((N+5)/2))]):locs:= select(t -> Primes[t+2]-Primes[t]=6, [$1..nops(Primes)-2]):
Primes[locs]; # Robert Israel, Apr 30 2015

ptrsQ[n_]:=PrimeQ[n+6]&&(PrimeQ[n+2]||PrimeQ[n+4])
Select[Prime[Range[400]],ptrsQ]  (* Harvey P. Dale, Mar 08 2011 *)

p=2;q=3;forprime(r=5,1e4,if(r-p==6,print1(p", "));p=q;q=r) \\ Charles R Greathouse IV, May 07 2012

a(n) = A098415(n) - 6. - Zak Seidov, Apr 30 2015

A320701 Indices of primes followed by a gap (distance to next larger prime) of 6.

Original entry on oeis.org

9, 11, 15, 16, 18, 21, 23, 32, 36, 37, 39, 40, 51, 54, 55, 56, 58, 67, 71, 73, 74, 76, 84, 86, 96, 100, 102, 103, 105, 107, 108, 110, 111, 118, 119, 123, 129, 130, 133, 160, 161, 164, 165, 167, 170, 174, 179, 184, 185, 187, 188, 194, 195, 199, 200, 202, 208, 210, 216, 218, 219, 227, 231

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031924.

Subsequence of indices of sexy primes A023201.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Equals A000720 o A031924.
Row 3 of A174349.
Indices of 6's in A001223.
Cf. A029707, A029709, A320702, A320703, ..., A320720 (analog for gaps 2, 4, 8, 10, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

Position[Differences[Prime[Range[250]]],6]//Flatten (* Harvey P. Dale, Oct 13 2022 *)

A(N=100,g=6,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031924(n)).

A320701 = { i > 0 | prime(i+1) = prime(i) + 6 } = A001223^(-1)({6}).

A023271 Primes p such that p, p+6, p+12, p+18 are all primes.

Original entry on oeis.org

5, 11, 41, 61, 251, 601, 641, 1091, 1481, 1601, 1741, 1861, 2371, 2671, 3301, 3911, 4001, 5101, 5381, 5431, 5641, 6311, 6361, 9461, 11821, 12101, 12641, 13451, 14621, 14741, 15791, 15901, 17471, 18211, 19471, 20341, 21481, 23321, 24091, 26171, 26681

Offset: 1

David W. Wilson

nonn

Smallest member of a "sexy" prime quadruple.
For n > 1, a(n) ends in 1. - Robert Israel, Jul 16 2015
The only sexy prime quintuple corresponding to (p, p+6, p+12, p+18, p+24) starts with a(1) = 5, so this quintuple is (5, 11, 17, 23, 29) (see Wikipedia link and A206039). - Bernard Schott, Mar 10 2023

Matt C. Anderson and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..100 from Matt C. Anderson)
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]
Wikipedia, Sexy prime.

Cf. A023201, A046117, A046122, A046123, A046124, A206039.

[p: p in PrimesInInterval(2, 1000000) | forall{i: i in [ 6, 12, 18] | IsPrime(p+i)}]; // Vincenzo Librandi, Jul 15 2015

for a to 2*10^5 do
if `and`(isprime(a), isprime(a+6), isprime(a+12), isprime(a+18))
then print(a);
end if;
end do;
# code produces 109 primes
# Matt C. Anderson, Jul 15 2015

Select[Prime[Range[1000]], PrimeQ[# + 6] && PrimeQ[# + 12] && PrimeQ[# + 18] &] (* Vincenzo Librandi, Jul 15 2015 *)
(* The following program uses the AllTrue function from Mathematica version 10 *) Select[Prime[Range[3000]], AllTrue[# + {6, 12, 18}, PrimeQ] &] (* Harvey P. Dale, Jun 06 2017 *)

main(size)=my(v=vector(size),i,r=1,p);for(i=1,size,while(1,p=prime(r);if(isprime(p+6)&&isprime(p+12)&&isprime(p+18),v[i]=p;r++;break,r++))); v \\ Anders Hellström, Jul 16 2015

Edited by N. J. A. Sloane, Aug 04 2009 following a suggestion from Daniel Forgues

A006489 Numbers k such that k-6, k, and k+6 are primes.

Original entry on oeis.org

11, 13, 17, 23, 37, 47, 53, 67, 73, 103, 107, 157, 173, 233, 257, 263, 277, 353, 373, 563, 593, 607, 613, 647, 653, 733, 947, 977, 1097, 1103, 1123, 1187, 1223, 1283, 1297, 1367, 1433, 1453, 1487, 1493, 1607, 1613, 1663, 1747, 1753, 1783, 1867, 1873

Offset: 1

N. J. A. Sloane

A006562 without the first term 5 is a subsequence. - Zak Seidov, Apr 19 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Joseph Hirsch, Prime Triplets, Mathematics Teacher, 62 (1969), 467ff (see p. 471).
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979

Subsequence of A023201.

Cf. A006562.

a006489 n = a006489_list !! (n-1)
a006489_list = filter
   ((== 1) . a010051 . (subtract 6)) $ dropWhile (<= 6) a023201_list
-- Reinhard Zumkeller, Feb 25 2013

Select[Prime[Range[300]],And@@PrimeQ[#+{6,-6}]&] (* Harvey P. Dale, May 21 2012 *)

is(n)=isprime(n-6) && isprime(n) && isprime(n+6) \\ Charles R Greathouse IV, Feb 07 2017

More terms from James Sellers, Dec 24 1999

cp's OEIS Frontend

A172295 Partial sums of A023201.

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A338812 Smaller term of a pair of sexy primes (A023201) such that the distance to next pair (A227346) is a square.

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A255472 Number of decompositions of 2n into sums of two primes p <= q such that one or both p and q are elements of A023201.

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A022004 Initial members of prime triples (p, p+2, p+6).

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A046117 Primes p such that p-6 is also prime. (Upper of a pair of sexy primes.)

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A031924 Primes followed by a gap of 6, i.e., next prime is p + 6.

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A007529 Prime triples: p; p+2 or p+4; p+6 all prime.

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