A080356 -id:A080356 - cp's OEIS frontend

A143738 Number of twin primes between n and n^2. Only smaller of twins (terms of A001359) are counted.

0, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 8, 9, 11, 12, 14, 16, 16, 17, 17, 19, 20, 21, 22, 24, 26, 26, 26, 29, 30, 30, 31, 34, 36, 36, 39, 41, 42, 45, 45, 48, 49, 50, 53, 55, 59, 61, 63, 66, 66, 68, 70, 74, 74, 76, 77, 78, 83, 87, 90, 91, 93, 96, 100

Offset: 1

Alen Skugor (askugor(AT)gmail.com), Aug 30 2008

nonn

			For n = 6, between 6 and 36 the smaller of twin pairs are {11, 17, 29}, so a(6) = 3.

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

Cf. A001359, A080356.

[#[p:p in PrimesInInterval(n,n^2)|IsPrime(p+2)]:n in [1..80]]; // Marius A. Burtea, Dec 20 2019

Table[Count[Table[PrimeQ[j] && PrimeQ[j + 2], {j, n, n*n}], True], {n, 1, 100}]

A080357 Smallest x such that between x and 2x exactly n twin prime numbers can be found.

Original entry on oeis.org

2, 3, 36, 54, 90, 96, 99, 135, 174, 330, 411, 414, 516, 531, 546, 651, 714, 726, 741, 744, 804, 936, 939, 966, 975, 999, 1014, 1071, 1134, 1170, 1191, 1365, 1395, 1401, 1650, 1665, 1686, 1695, 1764, 1770, 1779, 1836, 1926, 2010, 2025, 2064, 2079, 2136, 2211

Offset: 1

Labos Elemer, Feb 19 2003

nonn

			n=7: first seven twin primes counted between 99 and 198: {101,107,137,149,179,191,197}.

Cf. A001359, A080356.

f[x_] := Count[Table[PrimeQ[j]&&PrimeQ[j+2], {j, x, 2*x}], True] t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10000}]; t

a(n)=Min[x; A080356(x)=n]

A344117 Number of twin prime pairs in the range (6n + 1, 6(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 4, 4, 3, 3, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5

Offset: 1

Ya-Ping Lu, Jun 24 2021

nonn

Conjecture: a(n) >= 1.

Cf. A071538, A079629, A080356, A237840, A332086, A337788.

from sympy import isprime
def istwin(m): return 1 if isprime(6*m-1)*isprime(6*m+1) == 1 else 0
ct1 = 0
for n in range(1, 100):
    ct1 += istwin(n); ct = 0
    for m in range (n + 1, n + ct1 + 1): ct += istwin(m)
    print(ct)

cp's OEIS Frontend

A143738 Number of twin primes between n and n^2. Only smaller of twins (terms of A001359) are counted.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A080357 Smallest x such that between x and 2x exactly n twin prime numbers can be found.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A344117 Number of twin prime pairs in the range (6n + 1, 6(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

cp's OEIS Frontend

A143738 Number of twin primes between n and n^2. Only smaller of twins (terms of A001359) are counted.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A080357 Smallest x such that between x and 2x exactly n twin prime numbers can be found.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A344117 Number of twin prime pairs in the range (6*n + 1, 6*(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A344117 Number of twin prime pairs in the range (6n + 1, 6(n + m) + 1], where m is the number of twin prime pairs, 6*k +- 1 for k = 1, 2, ..., n.