A273257 -id:A273257 - cp's OEIS frontend

A308344 a(n) = (A001359(n+1)^2 - 1)/24, where A001359 = lesser of twin primes; or: pentagonal numbers (A000326) whose indices are twin ranks (A002822).

1, 5, 12, 35, 70, 145, 210, 425, 477, 782, 925, 1335, 1520, 1617, 2147, 2380, 3015, 3290, 4030, 5017, 7315, 7740, 8855, 11310, 13490, 14950, 15862, 17120, 18095, 27270, 28085, 28497, 30602, 32340, 43265, 44290, 45850, 46905, 49595, 55200, 62935, 67947, 69230, 70525

Offset: 1

M. F. Hasler and A. Dinculescu, Jul 04 2019

nonn

Subsequence of A024702 which considers all primes rather than only twins.
This sequence seems to play an important role in studying the twin prime conjecture; see also A057767, A273257, and related.
Dinculescu calls the numbers M(j) = (prime(j)^2 - 1)/6 "basic numbers", and [M(j), M(j+1)] a "twin interval" when j is the index of a twin prime. He notes that the length of such an interval equals four times the corresponding twin rank k(j) = (prime(j) + prime(j+1))/6, see near eq.(3.3) in the 2018 paper.

			Sequence A001359 = {3, 5, 11, 17, 29, ...} lists the lesser members of pairs of twin primes, (3, 5), (5, 7), (11, 13), (17, 19), ...
We ignore the first and start with the second pair, (5, 7). We have (5^2 - 1)/24 = 1 = a(1).
Next comes the pair (11, 13), whence (11^2 - 1)/24 = 120/24 = 5 = a(2), etc.

M. F. Hasler, Table of n, a(n) for n = 1..10000
A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Cf. A000326, A002822, A001359, A024702.

(#^2-1)/24&/@Rest[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]] == 2&][[All,1]]] (* Harvey P. Dale, Sep 05 2020 *)

PARI
```
a(n)=A000326(A002822(n))
```

a(n)=(A001359(n+1)^2-1)/24 \\ or implemented as follows:
p=0;forprime(q=5,oo,p+2==q&&print1(p^2\24",");p=q)

a(n) = (A001359(n+1)^2 - 1)/24 = A000326(A002822(n)).

A308777 Number of twin primes between p and p^2 (inclusive) where p is the n-th prime.

Original entry on oeis.org

1, 3, 6, 9, 16, 19, 32, 35, 42, 58, 61, 82, 96, 101, 122, 148, 174, 183, 220, 242, 247, 276, 304, 332, 374, 404, 417, 436, 447, 468, 552, 576, 630, 641, 730, 749, 788, 822, 864, 910, 960, 985, 1082, 1095, 1134, 1149, 1252, 1370, 1416, 1433, 1464, 1528, 1545, 1636, 1702

Offset: 1

Michel Marcus, Jun 24 2019

nonn

Similar sequences given in cross-references have further information and references; in particular A273257 has much more efficient PARI code. - M. F. Hasler, Jun 27 2019

			There is a single twin prime (3) between 2 and 4, so a(1) = 1.
There are 3 twin primes (3, 5 and 7) between 3 and 9, so a(2) = 3.

Chai Wah Wu, Table of n, a(n) for n = 1..7944
Jon S. Birdsey, Geza Schay, A Sieve for Twin Primes, arXiv:1906.09220 [math.NT], 2019.

Cf. A001097 (twin primes), A054272, A057767 (twin pairs between p(n)^2 and p(n+1)^2), A088019.

Cf. A143738 (twin primes between n and n^2), A273257 (twin pairs between prime(n) and prime(n)^2).

a:= n-> (p-> add(`if`(isprime(j) and (isprime(j-2) or
        isprime(j+2)), 1, 0), j=p..p^2))(ithprime(n)):
seq(a(n), n=1..55);  # Alois P. Heinz, Jun 25 2019

a[n_] := With[{p = Prime[n]}, Sum[Boole[PrimeQ[k] && (PrimeQ[k-2] || PrimeQ[k+2])], {k, p, p^2}]];
Array[a, 55] (* Jean-François Alcover, Feb 29 2020 *)

a(n) = my(p=prime(n)); sum(k=p, p^2, isprime(k) && (isprime(k-2) || isprime(k+2)));

from sympy import prime, prevprime, nextprime
def A308777(n):
    if n == 1:
        return 1
    c, p = 0, prime(n)
    p2, x = p**2, [prevprime(p), p , nextprime(p)]
    while x[1] <= p2:
        if x[1] - x[0] == 2 or x[2] - x[1] == 2:
            c += 1
        x = x[1:] + [nextprime(x[2])]
    return c # Chai Wah Wu, Jun 25 2019

cp's OEIS Frontend

A308344 a(n) = (A001359(n+1)^2 - 1)/24, where A001359 = lesser of twin primes; or: pentagonal numbers (A000326) whose indices are twin ranks (A002822).

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A308777 Number of twin primes between p and p^2 (inclusive) where p is the n-th prime.

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