A071538 -id:A071538 - cp's OEIS frontend

A001359 Lesser of twin primes.

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607

Offset: 1

N. J. A. Sloane

Also, solutions to phi(n + 2) = sigma(n). - Conjectured by Jud McCranie, Jan 03 2001; proved by Reinhard Zumkeller, Dec 05 2002
The set of primes for which the weight as defined in A117078 is 3 gives this sequence except for the initial 3. - Rémi Eismann, Feb 15 2007
The set of lesser of twin primes larger than three is a proper subset of the set of primes of the form 3n - 1 (A003627). - Paul Muljadi, Jun 05 2008
It is conjectured that A113910(n+4) = a(n+2) for all n. - Creighton Dement, Jan 15 2009
I would like to conjecture that if f(x) is a series whose terms are x^n, where n represents the terms of sequence A001359, and if we inspect {f(x)}^5, the conjecture is that every term of the expansion, say a_n * x^n, where n is odd and at least equal to 15, has a_n >= 1. This is not true for {f(x)}^k, k = 1, 2, 3 or 4, but appears to be true for k >= 5. - Paul Bruckman (pbruckman(AT)hotmail.com), Feb 03 2009
A164292(a(n)) = 1; A010051(a(n) - 2) = 0 for n > 1. - Reinhard Zumkeller, Mar 29 2010
From Jonathan Sondow, May 22 2010: (Start)
About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes A104272 < 19000 are the lesser of twin primes.
About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan primes.
A reason for the jumps is in Section 7 of "Ramanujan primes and Bertrand's postulate" and in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps". (End)
Primes generated by sequence A040976. - Odimar Fabeny, Jul 12 2010
Primes of the form 2*n - 3 with 2*n - 1 prime n > 2. Primes of the form (n^2 - (n-2)^2)/2 - 1 with (n^2 - (n-2)^2)/2 + 1 prime so sum of two consecutive odd numbers/2 - 1. - Pierre CAMI, Jan 02 2012
Conjecture: For any integers n >= m > 0, there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) (i.e., (a(m), ..., a(n)) in base b) is prime; moreover, when m = 1 there is such an integer b < (n+6)^2. - Zhi-Wei Sun, Mar 26 2013
Except for the initial 3, all terms are congruent to 5 mod 6. One consequence of this is that no term of this sequence appears in A030459. - Alonso del Arte, May 11 2013
Aside from the first term, all terms have digital root 2, 5, or 8. - J. W. Helkenberg, Jul 24 2013
The sequence provides all solutions to the generalized Winkler conjecture (A051451) aside from all multiples of 6. Specifically, these solutions start from n = 3 as a(n) - 3. This gives 8, 14, 26, 38, 56, ... An example from the conjecture is solution 38 from twin prime pairs (3, 5), (41, 43). - Bill McEachen, May 16 2014
Conjecture: a(n)^(1/n) is a strictly decreasing function of n. Namely a(n+1)^(1/(n+1)) < a(n)^(1/n) for all n. This conjecture is true for all a(n) <= 1121784847637957. - Jahangeer Kholdi and Farideh Firoozbakht, Nov 21 2014
a(n) are the only primes, p(j), such that (p(j+m) - p(j)) divides (p(j+m) + p(j)) for some m > 0, where p(j) = A000040(j). For all such cases m=1. It is easy to prove, for j > 1, the only common factor of (p(j+m) - p(j)) and (p(j+m) + p(j)) is 2, and there are no common factors if j = 1. Thus, p(j) and p(j+m) are twin primes. Also see A067829 which includes the prime 3. - Richard R. Forberg, Mar 25 2015
Primes prime(k) such that prime(k)! == 1 (mod prime(k+1)) with the exception of prime(991) = 7841 and other unknown primes prime(k) for which (prime(k)+1)*(prime(k)+2)*...*(prime(k+1)-2) == 1 (mod prime(k+1)) where prime(k+1) - prime(k) > 2. - Thomas Ordowski and Robert Israel, Jul 16 2016
For the twin prime criterion of Clement see the link. In Ribenboim, pp. 259-260 a more detailed proof is given. - Wolfdieter Lang, Oct 11 2017
Conjecture: Half of the twin prime pairs can be expressed as 8n + M where M > 8n and each value of M is a distinct composite integer with no more than two prime factors. For example, when n=1, M=21 as 8 + 21 = 29, the lesser of a twin prime pair. - Martin Michael Musatov, Dec 14 2017
For a discussion of bias in the distribution of twin primes, see my article on the Vixra web site. - Waldemar Puszkarz, May 08 2018
Since 2^p == 2 (mod p) (Fermat's little theorem), these are primes p such that 2^p == q (mod p), where q is the next prime after p. - Thomas Ordowski, Oct 29 2019, edited by M. F. Hasler, Nov 14 2019
The yet unproved "Twin Prime Conjecture" states that this sequence is infinite. - M. F. Hasler, Nov 14 2019
Lesser of the twin primes are the set of elements that occur in both A162566, A275697. Proof: A prime p will only have integer solutions to both (p+1)/g(p) and (p-1)/g(p) when p is the lesser of a twin prime, where g(p) is the gap between p and the next prime, because gcd(p+1,p-1) = 2. - Ryan Bresler, Feb 14 2021
From Lorenzo Sauras Altuzarra, Dec 21 2021: (Start)
J. A. Hervás Contreras observed the subsequence 11, 311, 18311, 1518311, 421518311... (see the links), which led me to conjecture the following statements.
I. If i is an integer greater than 2, then there exist positive integers j and k such that a(j) equals the concatenation of 3k and a(i).
II. If k is a positive integer, then there exist positive integers i and j such that a(j) equals the concatenation of 3k and a(i).
III. If i, j, and r are positive integers such that i > 2 and a(j) equals the concatenation of r and a(i), then 3 divides r. (End)

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 81.
Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1996, pp. 259-260.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 192-197.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 111-112.

Chris K. Caldwell, Table of n, a(n) for n = 1..100000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Abhinav Aggarwal, Zekun Xu, Oluwaseyi Feyisetan, and Nathanael Teissier, On Primes, Log-Loss Scores and (No) Privacy, arXiv:2009.08559 [cs.LG], 2020.
Chris K. Caldwell, First 100000 Twin Primes
Chris K. Caldwell, Twin Primes
Chris K. Caldwell, Largest known twin primes
Chris K. Caldwell, Twin primes
Chris K. Caldwell, The prime pages
P. A. Clement, Congruences for sets of primes, American Mathematical Monthly, vol. 56,1 (1949), 23-25.
Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004; Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
José Antonio Hervás Contreras, ¿Nueva propiedad de los primos gemelos?
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant [Local copy, pdf only]
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Waldemar Puszkarz, Statistical Bias in the Distribution of Prime Pairs and Isolated Primes, vixra:1804.0416 (2018).
Fred Richman, Generating primes by the sieve of Eratosthenes
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
P. Shiu, A Diophantine Property Associated with Prime Twins, Experimental mathematics 14 (1) (2005).
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010; Amer. Math. Monthly, 116 (2009) 630-635.
Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
Jonathan Sondow and Emmanuel Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Terence Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Eric Weisstein's World of Mathematics, Twin Primes
Index entries for primes, gaps between

Subsequence of A003627.
Cf. A001223, A006512 (greater of twin primes), A014574, A001097, A077800, A002822, A040040, A054735, A067829, A082496, A088328, A117078, A117563, A074822, A071538, A007508, A146214, A350246, A350247.
Cf. A104272 (Ramanujan primes), A178127 (lesser of twin Ramanujan primes), A178128 (lesser of twin primes if it is a Ramanujan prime).
Cf. A010051, A000040.

a001359 n = a001359_list !! (n-1)
a001359_list = filter ((== 1) . a010051' . (+ 2)) a000040_list
-- Reinhard Zumkeller, Feb 10 2015

[n: n in PrimesUpTo(1610) | IsPrime(n+2)];  // Bruno Berselli, Feb 28 2011

select(k->isprime(k+2),select(isprime,[$1..1616])); # Peter Luschny, Jul 21 2009
A001359 := proc(n)
   option remember;
   if n = 1
      then 3;
   else
      p := nextprime(procname(n-1)) ;
      while not isprime(p+2) do
         p := nextprime(p) ;
      end do:
      p ;
   end if;
end proc: # R. J. Mathar, Sep 03 2011

Select[Prime[Range[253]], PrimeQ[# + 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
a[n_] := a[n] = (p = NextPrime[a[n - 1]]; While[!PrimeQ[p + 2], p = NextPrime[p]]; p); a[1] = 3; Table[a[n], {n, 51}]  (* Jean-François Alcover, Dec 13 2011, after R. J. Mathar *)
nextLesserTwinPrime[p_Integer] := Block[{q = p + 2}, While[NextPrime@ q - q > 2, q = NextPrime@ q]; q]; NestList[nextLesserTwinPrime@# &, 3, 50] (* Robert G. Wilson v, May 20 2014 *)
Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&][[All,1]] (* Harvey P. Dale, Jan 04 2021 *)
q = Drop[Prepend[p = Prime[Range[100]], 2], -1];
Flatten[q[[#]] & /@ Position[p - q, 2]] (* Horst H. Manninger, Mar 28 2021 *)

A001359(n,p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0,); p-2}
/* The following gives a reasonably good estimate for any value of n from 1 to infinity; compare to A146214. */
A001359est(n) = solve( x=1,5*n^2/log(n+1), 1.320323631693739*intnum(t=2.02,x+1/x,1/log(t)^2)-log(x) +.5 - n)
/* The constant is A114907; the expression in front of +.5 is an estimate for A071538(x) */ \\  M. F. Hasler, Dec 10 2008

from sympy import primerange, isprime
print([n for n in primerange(1, 2001) if isprime(n + 2)]) # Indranil Ghosh, Jul 20 2017

a(n) = A077800(2n-1).
A001359 = { n | A071538(n-1) = A071538(n)-1 }; A071538(A001359(n)) = n. - M. F. Hasler, Dec 10 2008
A001359 = { prime(n) : A069830(n) = A087454(n) }. - Juri-Stepan Gerasimov, Aug 23 2011
a(n) = prime(A029707(n)). - R. J. Mathar, Feb 19 2017

A067611 Numbers of the form 6xy +- x +- y, where x, y are positive integers.

Original entry on oeis.org

4, 6, 8, 9, 11, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 29, 31, 34, 35, 36, 37, 39, 41, 42, 43, 44, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 94

Offset: 1

Jon Perry, Feb 01 2002

nonn

Equivalently, numbers n such that either 6n-1 or 6n+1 is composite (or both are).
Numbers k such that 36*k^2 - 1 is not a product of twin primes. - Artur Jasinski, Dec 12 2007
Apart from initial zero, union of A046953 and A046954. - Reinhard Zumkeller, Jul 13 2014
From Bob Selcoe, Nov 18 2014: (Start)
Complementary sequence to A002822.
For all k >= 1, a(n) are the only positive numbers congruent to the following residue classes:
f == k (mod 6k+-1);
g == (5k-1) (mod 6k-1);
h == (5k+1) (mod 6k+1).
All numbers in classes g and h will be in this sequence; for class f, the quotient must be >= 1.
When determining which numbers are contained in this sequence, it is only necessary to evaluate f, g and h when the moduli are prime and the dividends are >= 2*k*(3*k - 1) (i.e., A033579(k)).
(End)
From Jason Kimberley, Oct 14 2015: (Start)
Numbers n such that A001222(A136017(n)) > 2.
The disjoint union of A060461, A121763, and A121765.
(End)
From Ralf Steiner, Aug 08 2018 (Start)
Conjecture 1: With u(k) = floor(k(k + 1)/4) one has A071538(a(u(k))*6) = a(u(k)) - u(k) + 1, for  k >= 2 (u > 1).
Conjecture 2: In the interval [T(k-1)+1, T(k)], with T(k) = A000217(k), k >= 2, there exists at least one number that is not a member of the present sequence. (End)
Also: numbers of the form n*p +- round(p/6) with some positive integer n and prime p >= 5. [Proof available on demand.] - M. F. Hasler, Jun 25 2019

			4 = 6ab - a - b with a = 1, b = 1.
6 = 6ab + a - b or 6ab - a + b with a = 1, b = 1.
5 cannot be obtained by any values of a and b in 6ab - a - b, 6ab - a + b, 6ab + a - b or 6ab + a + b.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Balestrieri, An Equivalent Problem To The Twin Prime Conjecture, arXiv:1106.6050 [math.GM], 2011.

Cf. A002822, A010051, A037074, A046953, A046954, A060461, A070043, A070799, A121763, A121765, A136017, A136050, A071538 (pi_2).

Cf. A323674 (numbers 6xy +- x +- y including repetitions). - Sally Myers Moite, Jan 27 2019

Filtered([1..120], k-> not IsPrime(6*k-1) or not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019

a067611 n = a067611_list !! (n-1)
a067611_list = map (`div` 6) $
   filter (\x -> a010051' (x-1) == 0 || a010051' (x+1) == 0) [6,12..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [1..100] | not IsPrime(6*n-1) or not IsPrime(6*n+1)]; // Vincenzo Librandi, Nov 19 2014

filter:= n -> not isprime(6*n+1) or not isprime(6*n-1):
select(filter, [$1..1000]); # Robert Israel, Nov 18 2014

Select[Range[100], !PrimeQ[6# - 1] || !PrimeQ[6# + 1] &]
Select[Range[100],AnyTrue[6#+{1,-1},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 05 2019 *)

for(n=1, 1e2, if(!isprime(6*n+1) || !isprime(6*n-1), print1(n", "))) \\ Altug Alkan, Nov 10 2015

[n for n in (1..120) if not is_prime(6*n-1) or not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019

Edited by Robert G. Wilson v, Feb 05 2002

Edited by Dean Hickerson, May 07 2002

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

A079629 Number of twin prime pairs between p^2 and q^2 where (p,q) is the n-th twin prime pair.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 5, 3, 7, 6, 6, 10, 13, 7, 8, 9, 9, 7, 12, 18, 14, 24, 19, 10, 21, 21, 20, 20, 19, 22, 19, 24, 24, 27, 25, 30, 27, 23, 34, 29, 21, 35, 38, 30, 32, 30, 33, 36, 33, 30

Offset: 1

Paul Boddington, Jan 30 2003

Conjecturally a(n) is always positive. It seems that a(n) might tend to infinity.

			a(3)=2 because the third twin prime pair is (11,13) and there are 2 twin prime pairs between 121 and 169, namely (137,139) and (149,151).

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

Cf. A001359, A006512, A057767, A071538.

Cf. A137859, A137860.

ntpp[{a_,b_}]:=Count[Differences[Select[Range[a^2+1,b^2],PrimeQ]],2]; ntpp/@ Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Oct 04 2019 *)

a(n) = A071538(A006512(n)^2) - A071538(A001359(n)^2). - Reinhard Zumkeller, Feb 16 2008

A080356 Number of twin primes between n and 2n: a(n) = number of j in range n <= j <= 2*n such that j and j+2 are primes.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6

Offset: 1

Labos Elemer, Feb 19 2003

nonn

			n=36: between 36 and 72 the primes p such that (p,p+2) is a pair of twin primes are {41,59,71}, so a(36)=3.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A001359, A071538.

Table[Count[Table[PrimeQ[j]&&PrimeQ[j+2], {j, n, 2*n}], True], {n, 1, 1000}]

Definition clarified by N. J. A. Sloane, Aug 22 2021

A098424 Number of prime triples (p,q,r) <= n with p

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, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Convention: a prime triple is <= n iff its smallest member is <= n;

a(n) <= A098428(n).

			a(15) = #{(5,7,11),(7,11,13),(11,13,17),(13,17,19)} = 4.

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

Cf. A007529, A098414, A098415, A071538.

Cf. A010051, A000040.

a098424 n = length [(p,q,r) | p <- takeWhile (<= n) a000040_list,
            let r = p + 6, a010051 r == 1, q <- [p+1..r-1], a010051 q == 1]
-- Reinhard Zumkeller, Nov 15 2011

With[{pts=Select[Partition[Prime[Range[1200]],3,1],Last[#]-First[#] == 6&]}, Table[Count[pts,?(First[#]<=n&)],{n,110}]] (* _Harvey P. Dale, Nov 09 2011 *)

A098429 Number of cousin prime pairs (p, p+4) with p <= n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 07 2004

nonn

Convention: a prime pair is <= n iff its smallest member is <= n.

Except for (3, 7), there is only 1 pair congruence class for cousin primes, i.e. (+1, -1) (mod 6). [Daniel Forgues, Aug 05 2009]

			First cousin prime pairs: (3,7),(7,11),(13,17),(19,23), ...
therefore the sequence starts: 0 0 1 1 1 1 2 2 2 2 2 2 3 ...

Daniel Forgues, Table of n, a(n) for n=1..99996
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A023200, A046132, A098428, A071538.

Accumulate[Table[If[PrimeQ[i]&&PrimeQ[i+4],1,0],{i,1,100}]] (* Seiichi Kirikami, May 28 2017 *)

Edited by Daniel Forgues, Aug 01 2009

A082462 Let chi(k) = 1 if prime(k+1) - prime(k) = 2, = 0 otherwise; sequence gives a(n) = sum_{k <= n} chi(k).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21

Offset: 1

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Apr 26 2003

nonn

			n p D a(n) (p=prime(n), D = prime(n+1)-prime(n))
1 2 1 0
2 3 2 1
3 5 2 2
4 7 4 2
5 11 2 3
6 13 4 3

Cf. A000040, A001223.

a(n)=A071538(A000040(n)). [From R. J. Mathar, Oct 06 2008]

A071702 Number of twin prime pairs <= n of form (4k+3,4(k+1)+1), k>=0.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Jun 04 2002

nonn

As for A071538 the convention is followed that a twin prime pair is <= n if its smaller member is <= n.

If {a,b} is a twin prime pair (aHarvey P. Dale, Nov 12 2021

			a(60)=3, since (59,61) is included along with (3,5) and (11,13).

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

a(n) = A071538(n) - A071701(n), cf. A071698, A071699.

Accumulate[Table[If[AllTrue[{n,n+2},PrimeQ]&&Mod[n,4]==3,1,0],{n,100}]] (* Harvey P. Dale, Nov 12 2021 *)

A072746 Number of pairs of primes (p, q) such that, for some integer k, (p+q)/2 = 2^k, 2^(k-1) < p < q, and p <= n.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Reinhard Zumkeller, Jul 08 2002

nonn

Cf. A072742, A072743, A071538.

Name corrected by Jon E. Schoenfield, Jun 30 2021

cp's OEIS Frontend

A001359 Lesser of twin primes.

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A067611 Numbers of the form 6xy +- x +- y, where x, y are positive integers.

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

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A079629 Number of twin prime pairs between p^2 and q^2 where (p,q) is the n-th twin prime pair.

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A080356 Number of twin primes between n and 2n: a(n) = number of j in range n <= j <= 2*n such that j and j+2 are primes.

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A098424 Number of prime triples (p,q,r) <= n with p

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A098429 Number of cousin prime pairs (p, p+4) with p <= n.

A071702 Number of twin prime pairs <= n of form (4k+3,4(k+1)+1), k>=0.