A114429 -id:A114429 - cp's OEIS frontend

A003618 Largest n-digit prime.

7, 97, 997, 9973, 99991, 999983, 9999991, 99999989, 999999937, 9999999967, 99999999977, 999999999989, 9999999999971, 99999999999973, 999999999999989, 9999999999999937, 99999999999999997, 999999999999999989, 9999999999999999961, 99999999999999999989

Offset: 1

N. J. A. Sloane, Mira Bernstein

Since 10^n - 1 is always a multiple of 9, one could be tempted to think that 9 is the least frequently occurring least significant digit in terms of this sequence. - Alonso del Arte, Dec 03 2017

The occurrences of least significant digits in the first 8000 terms (see A033874) are 1: 2028, 3: 2032, 7: 2014, and 9: 1926. - Giovanni Resta, Mar 16 2020

			No power of 10 is prime.
9 = 3^2, 8 = 2^3 but 7 is prime, so a(1) = 7.
99 = 3^2 * 11 but 97 is prime, so a(2) = 97.
999 = 3^3 * 37 but 997 is prime, so a(3) = 997.
9999 = 3^2 * 11 * 101, 9997 = 13 * 769, 9995 = 5 * 1999, 9993 = 3 * 3331, 9991 = 97 * 103, ..., 9975 = 5^2 * 399, but 9973 is prime, so a(4) = 9973.

O'Hara, J. Rec. Math., 22 (1990), Table on page 278.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..1000 (terms 1..200 from T. D. Noe. _Jon E. Schoenfield_ verified that terms 1..200 are indeed primes, Feb 02 2009)
Eric Weisstein's World of Mathematics, Previous Prime
R. G. Wilson, v., Extract from letter to N. J. A. Sloane, May 20 1994, with annotated scanned copy of page 278 of O'Hara article.

Cf. A003617, A033874, A114429 (largest n-digit twin prime).

[PreviousPrime(10^n): n in [1..20]]; // Vincenzo Librandi, Sep 13 2016

a:= n-> prevprime(10^n):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 11 2021

NextPrime[10^Range[20], -1] (* Harvey P. Dale, Feb 03 2011 *)

a(n)=precprime(10^n) \\ Charles R Greathouse IV, Jul 19 2011

from sympy import prevprime
print([prevprime(10**n) for n in range(1, 21)]) # Michael S. Branicky, Feb 11 2021

More terms from Stefan Steinerberger, Apr 08 2006

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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A003618 Largest n-digit prime.

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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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Programs

PARI

Python