A001359 -id:A001359 - cp's OEIS frontend

A171727 The number of twin prime pairs in the interval (p^2,p*q), where (p,q) runs over the twin prime pairs (A001359(n),A006512(n)).

1, 1, 1, 1, 2, 2, 4, 1, 3, 2, 2, 4, 7, 3, 3, 5, 7, 4, 4, 7, 6, 11, 9, 5, 11, 9, 9, 11, 10, 11, 9, 11, 11, 12, 11, 12, 18, 12, 12, 16, 11, 16, 20, 14, 16, 15, 20, 16, 22, 13, 22, 16, 17, 21, 20, 20, 23, 22, 23, 20, 21, 21, 26, 20, 28, 24, 24, 23, 24, 25, 21, 24, 37, 27, 21, 28, 24, 31

Offset: 1

Jaspal Singh Cheema, Dec 16 2009

nonn

If you graph the order of the twin primes along the x-axis (i.e., first twin, second, third, ...) and the number of twins in the sequence given above along the y-axis, a clear pattern emerges. As you go farther along the x-axis, the number of twin primes, on average, within the interval increases. The pattern appears to be nonlinear. If one could prove that there's at least one twin prime within each interval, the twin prime conjecture would be proved since the n-th twin produces larger intervals with more twin primes. The evidence seems overwhelming.

			The first twin prime pair (3,5) corresponds to the interval (9,15), which contains one twin prime pair (11,13), so a(1) = 1.
The fifth twin prime pair (29,31) corresponds to the interval (841,899), which contains the twin prime pairs (857,859) and (881,883), so a(5) = 2.

C. C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Perseus Books, 1999.
J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Penguin Books Canada Ltd., 2004.
M. du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins Publishers Inc., 2004.

J. S. Cheema, Table of n, a(n) for n = 1..1044

Cf. A001359, A006512, A108570, A037074. - Charlie Neder, Feb 12 2019

{for(k=1, 300, if(prime(k+1)-prime(k)==2, my(c=0); forprime(m=prime(k)^2, prime(k)*prime(k+1), c+=isprime(m+2)); print1(c, ", ")))} \\ Zhandos Mambetaliyev, Mar 28 2021

Partially edited by Michel Marcus, Mar 19 2013

Edited by Charlie Neder, Feb 12 2019

A182483 a(n) is the least m such that A182482(m) = A001359(n), the n-th twin prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 4, 17, 9, 23, 25, 15, 8, 11, 19, 20, 45, 47, 13, 29, 14, 24, 77, 87, 95, 50, 103, 107, 22, 27, 137, 46, 143, 21, 34, 43, 175, 59, 91, 48, 41, 71, 215, 31, 44, 119, 121, 247, 62, 67, 54, 139, 283, 287, 149, 39, 313, 161, 65, 37, 169, 347, 116

Offset: 2

Vladimir Shevelev, May 01 2012

nonn

a(n) exists for every n>=2.

Ray Chandler, Table of n, a(n) for n = 2..10001

Cf. A182481, A182482, A001359, A006512, A014574, A001097, A077800, A002822.

t = Table[k = 0; While[p = 6*k*n - 1; ! (PrimeQ[p] && PrimeQ[p + 2]), k++]; p, {n, 1000}]; tp = Select[Prime[Range[1000]], PrimeQ[# + 2] &]; t2 = {}; found = True; n = 2; While[found, pos = Position[t, tp[[n]], 1, 1]; If[pos == {}, found = False, AppendTo[t2, pos[[1, 1]]]; n++]]; t2 (* T. D. Noe, May 02 2012 *)

A242917 Positions of smaller of twin primes in A001359 with index 4.

Original entry on oeis.org

7, 16, 20, 28, 30, 34, 35, 40, 49, 69, 74, 84, 89, 101, 105, 108, 133, 134, 142, 148, 154, 159, 169, 176, 182, 185, 194, 202, 213, 215, 220, 221, 225, 232, 235, 238, 251, 256, 261, 303, 310, 311, 314, 322, 323, 329, 330, 342, 343, 353, 354, 360, 382, 396, 398

Offset: 1

Vladimir Shevelev, May 26 2014

nonn

For the definition of the index of a twin prime pair, see the comment in A242767.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A001359, A242758, A242767, A242768, A242807, A242881, A242913.

A242767=Join[{1},Differences[Flatten[Position[Differences[Prime[Range[10^4]]],2]]]];
A242917=Flatten[Position[A242767,4]] (* Peter J. C. Moses, May 27 2014 *)

More terms from Peter J. C. Moses, May 26 2014

A292691 a(n) = C(A001359(n)), n >= 1, with C(n) = (4((n-1)! + 1) + n)/(n(n+2)) for n >= 2.

Original entry on oeis.org

1, 3, 101505, 259105757127, 1356566605613854774200240267, 1851197466245939272480116323530608949000567215

Offset: 1

Jaime Gómez, Sep 20 2017

nonn

Clement's criterion for twin primes is, for integers with n >= 2: n and n + 2 are both primes if and only if 4*((n-1)! + 1) + n == 0 (mod n*(n+2)). See the Clement and Ribenboim links. Like the criteron for primality using Theorem 81 of Hardy and Wright, p. 69, it "is of course quite useless as a practical test".
a(n) is an integer because of the necessary part of this twin prime criterion.
Thanks to Wolfdieter Lang for comments and helpful advice.

			a(2) = 3, because A001359(2) = 5 and C(5) = (4*(4! + 1) + 5)/(5*7) = 3.
a(2) = 3 because A014574(2) = 6 and delta(6) = (4*4! + 6 + 3)/35 = 3.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, 1979.
P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1996, pp. 259-260 (a proof of Clement's theorem).

Jaime Gómez, Table of n, a(n) for n = 1..23
P. A. Clement, Congruences for sets of primes, American Mathematical Monthly, vol. 56 (1949), pages 23-25.
L. Cong and Z. Li, On Wilson's Theorem and Polignac's Conjecture, arXiv:math/0408018 [math.NT], 2004.

Cf. A001359, A007619, A014574.

p1[1] = 3; p1[n_] := p1[n] = (p = NextPrime[p1[n-1]]; While[!PrimeQ[p + 2], p = NextPrime[p]]; p);
a[n_] := (4*((p1[n] - 1)! + 1) + p1[n])/(p1[n]*(p1[n] + 2));
Array[a, 6] (* Jean-François Alcover, Nov 04 2017 *)

c(n) = (4*(n - 2)! + n + 3) / (n^2 - 1);
lista(nn) = forprime(p=2, nn, if (isprime(p+2), print1(c(p+1), ", "));); \\ Michel Marcus, Sep 21 2017

# Python version 2.7
import math
from sympy import *
list = []
n = 3
l = 1   # parameter that indicates the desired length of the list
x = 1
while x <= l:
       y = (4*factorial(n-2))+n+3
       z = n**2 - 1
       if y % z == 0:
              print (y/z)
              list.append(y/z)
       n+=1
       x+=1

a(n) = (4*((p1(n)-1)! + 1) + p1(n))/(p1(n)*(p1(n) + 2)) with p1(n) = A001359(n), for n >= 1. See the name.
From Wilson's theorem (see Hardy and Wright, Theorem 80, p. 68), a(n) = (4*kp1(n) + 1)/(p1(n) + 2) with p1(n) = A000359(n) and kp1(n) = A007619(p1(n)).
a(n) = delta(A014574(n)) with delta(n) = (4*(n-2)!+ n + 3)/(n^2 - 1).
delta(n) ~ ((4*(n-2)^(n - 2)* sqrt(2*Pi*(n - 2))) / (e^(n - 2)*(n^2 - 1)))+((n + 3) / (n^2 - 1)) for large n-values (using Stirling's approximation for n!).

Edited by Wolfdieter Lang, Oct 25 2017

A342714 Decimal expansion of infinite sum of reciprocals of lesser twin primes, Sum_{n>=1} 1/A001359(n).

Original entry on oeis.org

1, 0, 5, 9, 0, 6, 4, 2, 6

Offset: 1

Artur Jasinski, Mar 19 2021

Alternative definition: infinite sum of reciprocals of primes whose distance to the next prime is equal to 2.

R. J. Mathar gave an estimate of 1.059064 for this constant in a comment at A209328. Dimitris Valianatos estimated the constant as 1.059064266555685... in a comment at A306539.

			Equals 1.05906426...

Cf. A006512, A065421, A077800, A160910, A209328, A209329, A306539.

Equals 1/3 + 1/5 + 1/11 + 1/17 + 1/29 + 1/41 + 1/59 + ...

Equals (A065421 + A306539)/2.

A350246 a(n) is the minimum positive integer k such that the concatenation of k, a(n-1), a(n-2), ..., a(2), and a(1) is the lesser of a pair of twin primes (i.e., a term of A001359), with a(1) = 11.

Original entry on oeis.org

11, 3, 18, 15, 42, 189, 306, 369, 6, 1176, 93, 963, 2202, 750, 408, 498, 267, 1875, 240, 2751, 798, 1929, 3402, 6162, 6195, 4953, 5004, 8130, 18591, 20019, 4461, 1851, 46866, 29232, 7206, 24807, 4644, 23307, 48528, 21594, 28236, 4353, 28212, 3003, 22611, 50760

Offset: 1

Lorenzo Sauras Altuzarra, Dec 21 2021

First observed by J. A. Hervás Contreras (see the links).

Every term (from the second on) is a multiple of 3.

			11, 311, 18311, 1518311, and 421518311 are terms of A001359.

Chai Wah Wu, Table of n, a(n) for n = 1..100
José Antonio Hervás Contreras, ¿Nueva propiedad de los primos gemelos?

Cf. A001359.

terms := proc(n)
   local i, j, p, q, L, M:
   i, L, M := 0, [11], [11]:
   while numelems(L) < n do
      i, j := i+1, 0:
      while 1 > 0 do
         j, p := j+1, M[numelems(M)]:
         q := parse(cat(j, p)):
         if isprime(q) and isprime(q+2) then
            L, M := [op(L), j], [op(M), q]:
            break: fi: od: od:
   L: end:

from itertools import count, islice
from sympy import isprime
def A350246_gen(): # generator of terms
    yield 11
    s = '11'
    while True:
        for k in count(3,3):
            t = str(k)
            m = int(t+s)
            if isprime(m) and isprime(m+2):
                yield k
                break
        s = t+s
A350246_list = list(islice(A350246_gen(),20)) # Chai Wah Wu, Jan 12 2022

A359640 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n odd prime factors, counted with multiplicity.

Original entry on oeis.org

307, 1999, 101527, 7146697, 272572999, 4809363523

Offset: 2

Hugo Pfoertner, Jan 16 2023

			a(2) = 307: 308 = 2^2*7*11, 309 = 3*103, 310 = 2*5*31, all have exactly 2 odd prime factors.

Cf. A001222, A001359, A087436, A359637, A359639.

a087436(n) = bigomega (n >> valuation (n, 2));
a359640(maxp) = {my(k=2, pp=5); forprime (p=7, maxp, my(mi=oo, ma=0); if (p-pp>2, for (j=pp+1, p-1, my(mo=a087436(j)); if (mo

A064409 Positive even numbers not of the form A001359(i) + A006512(j) for integers i and j.

Original entry on oeis.org

2, 4, 6, 14, 20, 26, 28, 32, 38, 40, 44, 50, 52, 56, 58, 62, 68, 70, 74, 80, 82, 86, 88, 92, 94, 96, 98, 100, 104, 110, 116, 118, 122, 124, 128, 130, 134, 136, 140, 146, 148, 152, 158, 160, 164, 166, 170, 172, 176, 178, 182, 188, 190, 194, 200

Offset: 1

Robert G. Wilson v, Sep 29 2001

nonn

This is different from A007534, which gives positive even numbers which are not the same of a pair of twin primes. The old definition of the present sequence was misleading. - N. J. A. Sloane, Feb 16 2024

			The lesser of the twin primes < 200 are 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197 and the greater of the twin primes < 200 are 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199. 20 is in the current sequence because no combination of any two numbers from each set just enumerated can be summed to make 20.

A.H.M. Smeets, Table of n, a(n) for n = 1..20177

Cf. A001359, A006512, A007534.

p = Select[ Range[ 200 ], PrimeQ[ # ] && PrimeQ[ # + 2 ] & ]; q = p + 2; Complement[ Table[ n, {n, 2, 200, 2} ], Union[ Flatten[ Table[ p[ [ i ] ] + q[ [ j ] ], {i, 1, 15}, {j, 1, 15} ] ] ] ]

A099728 Least number B such that (A001359(n) - B^2)^2 - B is also the lesser of larger twin primes, or 0 if no such B exists.

Original entry on oeis.org

380, 14, 5, 5, 365, 8, 5, 5, 14, 20, 5, 20, 8, 65, 8, 95, 35, 8, 14, 65, 20, 65, 8, 17, 350, 188, 5, 104, 98, 68, 35, 17, 158, 35, 92, 50, 62, 5, 26, 8, 8, 68, 233, 110, 5, 50, 23, 23, 8, 65, 59, 35, 14, 23, 35, 20, 47, 140, 5, 50, 14, 5, 44, 125, 386, 713, 23, 59, 44, 635, 98

Offset: 1

Ray G. Opao, Nov 07 2004

nonn

Conjecture: No term is zero.

			a(3) = 5 since A001359(3) = 11, 11 and 13 are twin primes, (11 - 5^2)^2 - 5 = 191, and 191 and 193 are also twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001359, A099742.

a135 := [] : f := fopen("b001359.txt",READ) : while nops(a135) < 200 do l := fscanf(f,"%d %d") : if l = [] then break : else a135 := [op(a135),l[2]] : fi ; od : for n from 1 to nops(a135) do a := op(n,a135) : B := 0 : while true do srch := (a-B^2)^2-B ; if isprime(srch) and isprime(srch+2) and srch > a then printf("%d, ",B) ; break ; fi ; B := B+1 : od : od: # R. J. Mathar, Aug 06 2007

f[p_] := Module[{b = 1}, While[(pb = (p - b^2)^2 - b) <= p || ! And @@ PrimeQ[pb + {0, 2}], b++]; b]; seq = {}; Do[If[And @@ PrimeQ[p + {0, 2}], AppendTo[seq, f[p]]], {p, 2, 3000}]; seq (* Amiram Eldar, Dec 30 2019 *)

Corrected and extended by R. J. Mathar, Aug 06 2007

Data corrected by Amiram Eldar, Dec 30 2019

A099742 Least number B such that (A001359(n)-B^2)^2+B is also the lesser of larger twin primes, or 0 if no such B exists.

Original entry on oeis.org

1, 1, 1, 7, 412, 7, 7, 133, 7, 7, 316, 7, 25, 10, 10, 10, 7, 7, 16, 10, 7, 25, 1, 7, 7, 100, 55, 7, 28, 940, 37, 148, 22, 16, 28, 67, 31, 82, 64, 4, 82, 445, 292, 310, 16, 1687, 13, 37, 43, 7, 58, 22, 31, 97, 70, 7, 22, 1, 19, 52, 58, 25, 1, 367, 4, 7, 4, 37, 55

Offset: 1

Ray G. Opao, Nov 09 2004

nonn

Conjecture: No term is zero.

			a(3) = 1 since A001359(3) = 11, 11 & 13 are twin primes, (11 - 1^2)^2 + 1 = 101, and 101 & 103 are also twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001359, A099728.

f[p_] := Module[{b = 1}, While[(pb = (p - b^2)^2 + b) <= p || ! And @@ PrimeQ[pb + {0, 2}], b++]; b]; seq = {}; Do[If[And @@ PrimeQ[p + {0, 2}], AppendTo[seq, f[p]]], {p, 2, 3000}]; seq (* Amiram Eldar, Dec 30 2019 *)

More terms from Amiram Eldar, Dec 30 2019

cp's OEIS Frontend

A171727 The number of twin prime pairs in the interval (p^2,p*q), where (p,q) runs over the twin prime pairs (A001359(n),A006512(n)).

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A182483 a(n) is the least m such that A182482(m) = A001359(n), the n-th twin prime.

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A242917 Positions of smaller of twin primes in A001359 with index 4.

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A292691 a(n) = C(A001359(n)), n >= 1, with C(n) = (4*((n-1)! + 1) + n)/(n*(n+2)) for n >= 2.

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A342714 Decimal expansion of infinite sum of reciprocals of lesser twin primes, Sum_{n>=1} 1/A001359(n).

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A350246 a(n) is the minimum positive integer k such that the concatenation of k, a(n-1), a(n-2), ..., a(2), and a(1) is the lesser of a pair of twin primes (i.e., a term of A001359), with a(1) = 11.

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A359640 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n odd prime factors, counted with multiplicity.

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A064409 Positive even numbers not of the form A001359(i) + A006512(j) for integers i and j.

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A292691 a(n) = C(A001359(n)), n >= 1, with C(n) = (4((n-1)! + 1) + n)/(n(n+2)) for n >= 2.