A092245 -id:A092245 - cp's OEIS frontend

A124001 Difference between first twin prime > 10^n and 10^n.

2, 1, 1, 19, 7, 151, 37, 139, 37, 7, 277, 817, 61, 1267, 97, 2371, 1549, 19, 619, 97, 391, 409, 649, 5527, 2731, 559, 949, 427, 601, 2797, 1681, 7189, 2449, 6751, 7597, 8419, 16879, 871, 5569, 10327, 16111, 2131, 6121, 23329, 5179, 4249, 2641, 2257, 3997

Offset: 0

Zak Seidov, Nov 01 2006

nonn

a(n) >= A033873(n) and a(n) = A033873(n) for n = 1, 2, 4, 9.

As N increases, the ratio (Sum_{n=1..N} a(n)/n^2)/N tends to 4. - Pierre CAMI, Jul 12 2013

			a(0) = 2 because 3 and 5 are twin primes and 3 - 10^0 = 2,
a(1) = 1 because 11 and 13 are twin primes and 11 - 10^1 = 1,
a(2) = 1 because 101 and 103 are twin primes and 101 - 10^2 = 1,
a(3) = 19 because 1019 and 1021 are twin primes and 1019 - 10^3 = 19, etc.

Robert G. Wilson v, Table of n, a(n) for n = 0..1250 (first 476 terms from _Pierre CAMI_).

Cf. A033873, A092245.

f[n_] := Block[{p = q = NextPrime[10^n]}, While[p + 2 != q, p = q; q = NextPrime@ q]; p - 10^n]; Array[f, 49, 0] (* Robert G. Wilson v, Nov 28 2015 *)
ftp[n_]:=Module[{p=NextPrime[n]},While[CompositeQ[p+2],p=NextPrime[p]];p-n]; Table[ftp[10^n],{n,0,50}] (* Harvey P. Dale, Oct 15 2019 *)

a(n) = A092245(n+1) - 10^n. - Robert G. Wilson v, Nov 28 2015

A092250 Lesser of the greatest twin prime pair with n digits.

Original entry on oeis.org

5, 71, 881, 9929, 99989, 999959, 9999971, 99999587, 999999191, 9999999701, 99999999761, 999999999959, 9999999998489, 99999999999971, 999999999997967, 9999999999999641, 99999999999998807, 999999999999998927

Offset: 1

Cino Hilliard, Feb 17 2004

Sum of reciprocals = 0.215331408...

Also the numerator of the largest prime-over-prime fraction less than 1 that is the ratio of two primes both less than 10^n. - Cino Hilliard, Feb 13 2006 [edited by Jon E. Schoenfield, Dec 01 2019]

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Robert G. Wilson v, Table of n, a(n) for n = 1..1250 (includes terms with more than 1000 digits)

Cf. A092245.

Cf. A114429(n) = a(n)+2: largest twin prime < 10^n.

Array[Block[{k = 10^# - 3}, While[! AllTrue[{k, k + 2}, PrimeQ], k -= 2]; k] &, 18]

lasttwpr(n) = { sr=0; for(m=0,n, c=0; forstep(x=10^(m+1)-1,10^m,-2, if(isprime(x)&& isprime(x-2),print1(x-2",");sr+=1./(x-2);break) ) ); print(); print(sr) }

apply( {A092250(n,p=10^n)=until(2==p-p=precprime(p-1),);p}, [1..22]) \\ avoids multiple isprime(): much faster! - M. F. Hasler, Jan 17 2022

from sympy import prevprime
def a(n):
    p = prevprime(10**n); pp = prevprime(p)
    while p - pp != 2: p, pp = pp, prevprime(pp)
    return pp
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Jan 17 2022

A240170 Larger of the greatest cousin prime pair with n digits.

Original entry on oeis.org

7, 83, 971, 9887, 99881, 999983, 9999401, 99999551, 999999761, 9999999707, 99999999947, 999999998867, 9999999999083, 99999999999467, 999999999997841, 9999999999997031, 99999999999998717, 999999999999999161, 9999999999999996587, 99999999999999999803

Offset: 1

Abhiram R Devesh, Aug 02 2014

The sum of the reciprocals converges to 0.156047....

It is only a (plausible) conjecture that this sequence is well-defined. See A152052. - N. J. A. Sloane, Aug 22 2014

Abhiram R Devesh, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A046132, A152052, A240167.
Analogous sequences with twin primes:
- A092245 Lesser of the first twin prime pair with n digits.
- A114429 Larger of the greatest twin prime pair with n digits.

a(n)=p=precprime(10^n);while(!isprime(p-4),p=precprime(p-1));return(p)
vector(50, n, a(n)) \\ Derek Orr, Aug 04 2014

import sympy
for i in range(1,100):
    a=(10**i)
    p=sympy.prevprime(a)
    while sympy.isprime(p-4)==False:
        p=sympy.prevprime(p)
    print(p)

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A124001 Difference between first twin prime > 10^n and 10^n.

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A092250 Lesser of the greatest twin prime pair with n digits.

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A240170 Larger of the greatest cousin prime pair with n digits.

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