A001359 -id:A001359 - cp's OEIS frontend

A052350 Least prime in A001359 (lesser of twin primes) such that the distance (A053319) to the next twin is 6*n.

5, 17, 41, 617, 71, 311, 2267, 521, 1877, 461, 1721, 347, 1151, 1787, 3581, 2141, 6449, 1319, 21377, 1487, 12251, 4799, 881, 23057, 659, 19541, 12377, 2381, 38747, 10529, 37361, 8627, 9041, 33827, 5879, 80231, 15359, 45821, 36107, 14627, 37991, 36527, 87251, 70997

Offset: 1

Labos Elemer, Mar 07 2000

nonn

Smallest distance (A052380) and also smallest possible increment of twin-distances is 6.
Primes may occur between p+2 and p+6n.
The prime a(n) determines a prime quadruple: [p, p+2, p+6n, p+6n+2] and a [2, 6n-2, 2] d-pattern.

			The first 3 terms (5, 17, 41) are followed by difference patterns as it is displayed: 5 by [2, 4, 2], 17 by [2, 4+6, 2], 41 by [2, 4+6+6, 2] determining prime quadruples: (5, 7, 11, 13), (17, 19, 29, 31) or (41, 43, 59, 61), respectively.
a(10) = 461 gives the quadruple [461, 463, 521 = 461+60, 523], and between 521 and 463, 7 primes occur.

Norman Luhn, Table of n, a(n) for n = 1..4624 (terms 1..4500 from Martin Raab).

Cf. A001359, A007530, A052380, A052381, A053319, A113274, A113275.

Cf. A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k += 6]; k];p = 5; t = Table[0, {50}]; Do[ q = NextLowerTwinPrim[p]; d = (q - p)/6; If[d < 51 && t[[d]] == 0, t[[d]] = p; Print[{d, p}]]; p = q, {n, 1500}]; t (* Robert G. Wilson v, Oct 28 2005 *)

list(len) = {my(s = vector(len), c = 0, p1 = 5, q1 = 0, q2, d); forprime(p2 = 7, , if(p2 == p1 + 2, q2 = p1; if(q1 > 0, d = (q2 - q1)/6; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 04 2025

Name corrected by Amiram Eldar, Mar 04 2025

A078859 Least positive residues (mod 210) representing those residue classes which can be the lesser of twin prime pairs (A001359).

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 167, 179, 191, 197, 209

Offset: 1

Labos Elemer, Dec 13 2002

Cf. A001359, A008364.

With[{n = 4}, Function[P, Join[Select[Prime@ Range@ n, NextPrime@ # == # + 2 &], Select[Partition[Select[Range[P + 1], CoprimeQ[#, P] &], 2, 1], Differences@ # == {2} &][[All, 1]]]]@ Product[Prime@ i, {i, n}]] (* Michael De Vlieger, May 15 2017 *)

Intersection[RRS(210), 2+RRS{210)]-2 and {3, 5}. RRS(210)=reduced residue system of 210=first 48=phi(210) terms of A008364; two additional term 3 and 5 are singular cases; 210k+r generates complete A001359 with suitable k and r taken from these 15+2 numbers.

A127552 Decimal expansion of the number 3.19644719338616871113868629540207517... having continued fraction expansion 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, ... (lesser of twin primes A001359).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jan 18 2007

Cf. A064442, A127549, A127550, A127551, A127553, A127555.

a = {}; Do[If[PrimeQ[Prime[n] + 2], AppendTo[a, Prime[n]]], {n, 2, 500}]; RealDigits[N[FromContinuedFraction[a], 100]][[1]]

a(100) corrected by Sean A. Irvine, Jul 09 2023

A097489 a(n) = product of first n terms of A001359.

Original entry on oeis.org

3, 15, 165, 2805, 81345, 3335145, 196773555, 13970922405, 1411063162905, 150983758430835, 20684774905024395, 3082031460848634855, 551683631491905639045, 105371573614953977057595, 20758200002145933480346215, 4712111400487126900038590805

Offset: 1

Cino Hilliard, Aug 24 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..279

Cf. A001359, A097492.

FoldList[Times,Select[Partition[Prime[Range[50]],2,1],#[[2]]-#[[1]]==2&][[All,1]]] (* Harvey P. Dale, Oct 09 2020 *)

twinl(n) = { local(c,x); c=0; x=1; while(c

Edited by Don Reble, Apr 16 2007

a(15)-a(16) from Amiram Eldar, Jul 07 2024

A242807 Smallest lesser of twin primes (A001359) with index n, or a(n)=0, if there are no such twin primes.

Original entry on oeis.org

3, 11, 29, 59, 137, 101, 2309, 1151, 521, 1427, 1229, 419, 5849, 3119, 5417, 10271, 1607, 9629, 1019, 809, 13217, 9239, 15581, 8819, 29021, 84059, 13679, 18911, 14867, 45119, 54401, 60647, 60089, 142589, 78137, 61979, 179381, 26681, 123377, 293861, 89519

Offset: 1

Vladimir Shevelev, May 23 2014

nonn

For definition of index of twin primes pair, see A242767.

We conjecture that, for every n, a(n)>0.

Cf. A001359, A242758, A242767.

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

A242881 Positions of lesser of twin primes in A001359 with index 2.

Original entry on oeis.org

3, 4, 10, 12, 14, 15, 23, 32, 33, 37, 39, 50, 58, 60, 64, 66, 82, 86, 90, 91, 93, 111, 112, 114, 128, 139, 143, 155, 157, 158, 162, 167, 171, 179, 190, 197, 198, 199, 207, 223, 226, 231, 241, 248, 255, 262, 270, 280, 282, 286, 293, 306, 313, 317, 318, 325, 327

Offset: 1

Vladimir Shevelev, May 25 2014

nonn

For the definition of the index of a twin primes 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.

More terms from Peter J. C. Moses, May 25 2014

A359636 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n distinct prime factors.

Original entry on oeis.org

7, 19, 643, 51427, 8083633, 1077940147, 75582271489, 34710483181813

Offset: 1

Hugo Pfoertner, Jan 12 2023

a(9) <= 76340177205657727, a(10) <= 225096507194749219819. - David A. Corneth, Jan 12 2023

			a(1) = 7: trivially, the 3 composites 8 = 2^3, 9 = 3^2, 10 = 2*5, have at least one distinct prime factor;
a(2) = 19: 20 = 2^2*5, 21 = 3*7, 22 = 2*11 all have 2 distinct prime factors;
a(3) = 643: 644 = 2^2*7*23, 645 = 3*5*43, 646 = 2*17*19, 647 is prime.

Nilotpal Kanti Sinha, Are there highly composite prime gaps? Question in mathoverflow, with an answer by Terry Tao, Jan 19 2022.

Cf. A001359, A075590, A185032.

a359636(maxp) = {my (k=1, pp=3); forprime (p=5, maxp, my(mi=oo); if (p-pp>2, for (j=pp+1, p-1, my(mo=omega(j)); if (mo=k, print1(pp,", "); k++)); pp=p)};
a359636(10^7)

a(8) from Martin Ehrenstein, Nov 03 2023

A242913 Positions of smaller of twin primes in A001359 with index 3.

Original entry on oeis.org

5, 6, 8, 17, 19, 29, 38, 44, 45, 46, 52, 54, 55, 65, 67, 71, 78, 80, 87, 95, 96, 97, 103, 106, 113, 119, 121, 124, 135, 136, 138, 152, 166, 187, 188, 191, 192, 208, 209, 212, 217, 237, 253, 254, 259, 269, 271, 275, 277, 288, 300, 308, 316, 320, 331, 349, 355

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.

A242767=Join[{1},Differences[Flatten[Position[Differences[Prime[Range[10^4]]],2]]]];
A242913=Flatten[Position[A242767,3]] (* Peter J. C. Moses, May 27 2014 *)

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

A247867 a(n) is the smallest prime in the interval [ksqrt(k), ksqrt(k+2)], where k = A001359(n), or a(n)=0 if there is no prime in this interval.

Original entry on oeis.org

0, 13, 37, 71, 157, 263, 457, 599, 1019, 1109, 1607, 1823, 2399, 2647, 2767, 3433, 3697, 4421, 4721, 5501, 6469, 8581, 8951, 9901, 11897, 13577, 14669, 15329, 16229, 16921, 23011, 23531, 23789, 25097, 26153, 32531, 33107, 33997, 34583, 36037, 39079, 43093

Offset: 1

Vladimir Shevelev, Sep 25 2014

nonn

The sequence is partly connected with conjecture in A247834. In turn, we conjecture that all terms a(n)>0 for n>1.

			For n=1, k=A001359(1)=3, we have the interval [3*sqrt(3), 3*sqrt(5)] = [5.1...,6.7...] which does not contain a prime. So, a(1)=0.
For n=2, k=5, we have the interval [5*sqrt(5), 5*sqrt(7)] = [11.1..., 13.2...] which contains only one prime: 13. So, a(2)=13.

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

Cf. A001359, A247834, A247835.

p:= 1: q:= 2: count:= 0:
while count < 100 do
  if q = p+2 then
    count:= count+1;
    r:= nextprime(floor(p*sqrt(p)));
    if r^2 < p^2*q then A[count]:= r
    else A[count]:= 0 fi;
  fi;
  p:= q; q:= nextprime(p);
od:
seq(A[i],i=1..100); # Robert Israel, Apr 08 2018

lista(nn) = {forprime(p=2, nn, if (isprime(q=p+2), pmin = nextprime(ceil(p*sqrt(p))); if (pmin <= floor(p*sqrt(q)), val = pmin, val = 0); print1(val, ", ");););} \\ Michel Marcus, Sep 25 2014

More terms from Michel Marcus, Sep 25 2014

A248891 Number of primes p such that p+2 is prime and A001359(n) < p < A001359(n)^(1+1/n).

Original entry on oeis.org

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

Offset: 1

Jahangeer Kholdi and Farideh Firoozbakht, Nov 22 2014

nonn

Conjecture: For every positive integer n, A001359(n+1)^(1/(n+1)) < A001359(n)^(1/n). Note that this conjecture is equivalent to " A001359 is infinite and for every n, A001359(n+1) < A001359(n)^(1+1/n). This implies for every n, a(n) is positive. See comment lines of the sequence A001359.

			Take n=1, A001359(1)=3, 3 < 5 < 3^(1+1/1)=9 hence a(1)=1.
Take n=6, A001359(6)=41, 41 < 59 < 71 < 41^(1+1/6)~76.13 hence a(6)=2.

Cf. A001359, A248901, A248902, A248903.

cp's OEIS Frontend

A052350 Least prime in A001359 (lesser of twin primes) such that the distance (A053319) to the next twin is 6*n.

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A078859 Least positive residues (mod 210) representing those residue classes which can be the lesser of twin prime pairs (A001359).

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A127552 Decimal expansion of the number 3.19644719338616871113868629540207517... having continued fraction expansion 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, ... (lesser of twin primes A001359).

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A097489 a(n) = product of first n terms of A001359.

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A242807 Smallest lesser of twin primes (A001359) with index n, or a(n)=0, if there are no such twin primes.

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A242881 Positions of lesser of twin primes in A001359 with index 2.

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A359636 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n distinct prime factors.

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A242913 Positions of smaller of twin primes in A001359 with index 3.

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A247867 a(n) is the smallest prime in the interval [k*sqrt(k), k*sqrt(k+2)], where k = A001359(n), or a(n)=0 if there is no prime in this interval.

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A247867 a(n) is the smallest prime in the interval [ksqrt(k), ksqrt(k+2)], where k = A001359(n), or a(n)=0 if there is no prime in this interval.