A213883 -id:A213883 - cp's OEIS frontend

A213884 For the smallest k >= 1, the smallest single-digit j such that (10^k-j)*10^n-1 is prime.

1, 4, 1, 2, 2, 5, 1, 2, 1, 2, 1, 4, 4, 5, 5, 1, 4, 7, 1, 4, 2, 4, 4, 1, 2, 8, 7, 4, 1, 1, 2, 1, 1, 4, 7, 4, 1, 1, 7, 4, 8, 2, 7, 4, 8, 8, 7, 2, 2, 1, 8, 2, 8, 5, 7, 1, 8, 4, 8, 1, 4, 1, 4, 7, 1, 2, 8, 2, 4, 1, 4, 8, 4, 5, 8, 2, 1, 2, 7, 7, 5, 1, 4, 8, 7, 4, 1, 4, 2, 2, 4, 5

Offset: 1

Pierre CAMI, Jun 29 2012

nonn

These j are the associated shifts to be paired with the k-values of A213883. There are no multiples of 3 here, as explained in A213883.

For the first 2200 values of n, there is always at least one pair (k,j) that delivers a prime with the conditions.

			j=1 associated with the prime 89, j=4 associated with 599, j=1 associated with 8999, j=2 with 79999 are the first 4 entries.

Pierre CAMI, Table of n, a(n) for n = 1..2200

Cf. A213883.

A213884 := proc(n)
    for k from 1 do
        for j from 0 to 9 do
            if isprime( (10^k-j)*10^n-1) then
                return j;
            end if;
        end do:
    end do:
    return 0 ;
end proc: # R. J. Mathar, Jul 20 2012

A228144 Smallest k > n such that j10^k + m10^n - 1 is a prime number for at least a pair {j,m} with 0 < j < 10 and 0 < m < 10.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 46, 47, 48, 49, 50, 51, 52, 53, 55, 55, 56, 57, 59, 59, 60, 61, 62, 63, 64, 66, 66, 67, 68, 70

Offset: 1

Pierre CAMI, Aug 14 2013

nonn

The prime numbers are the sum of a near repdigit number starting with the digit j followed by k digits 0 and a nearepdigit number starting with the digit (m-1) followed by n digits 9 for m>1, or for m=1 a repdigit number with n digits 9.
The first primes are :
109, 1399, 13999, 139999, 1199999, 16999999, 289999999, 2099999999, 10999999999, 239999999999, 1099999999999, 34999999999999, 349999999999999, 2399999999999999.
Conjecture: there is always at least one k for each n.

			1*10^1+1*10^2=109 prime so a(1)=2.

Pierre CAMI, Table of n, a(n) for n = 1..4000

Cf. A213790, A213883.

A213882 Numbers b such that there is at least one number c and one single-digit number d such that (10^c-d)10^b-1 and (10^c-d)10^b+1 are twin primes with 0 < c < 2*b.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 19, 21, 23, 26, 40, 43, 45, 52, 54, 55, 69, 77, 90, 99, 106, 128, 147, 176, 202, 267, 331, 458, 512, 555, 908, 942, 1004, 1123, 1374, 1386, 1467

Offset: 1

Pierre CAMI, Jun 26 2012

nonn

The single-digit number d is always 1, 4, or 7 in this format because otherwise one of (10^c-d)*10^b+-1 is a multiple of 3.
For one b there may be more than one matching (c,d).
The sequence of associated minimum c values starts: 1, 1, 1, 2, 3, 6, 1, 4, 2, 11, 9, 4, 7, 12, 9, 9, 42, 62, 5, 31, 2, 72, 88, 141, 119, 181, 6, 38, 164, 132, 53, 293, 150, 704, 557, 980, 952, 1596, 529, 2221, 200, 169, 1371,... and their associated d values are 4, 4, 1, 1, 1, 1, 7, 4, 7, 4, 1, 1, 4, 1, 7, 1, 4, 1, 7, 7, 7, 4, 1, 7, 1, 7, 4, 4, 4, 7, 1, 1, 7, 7, 4, 4, 4, 1, 1, 7, 7, 1, 1, ....

			(10^1-7)*10^1-1=29 prime 31 the twin prime so a(1)=1.
(10^1-4)*10^2-1=599 prime 601 the twin prime so a(2)=2.
(10^1-1)*10^3-1=8999 prime 9001 the twin prime so a(3)=3.
(10^2-1)*10^4-1=989999 prime 990001 twin prime so a(4)=4.
(10^3-1)*10^5-1=99899999 prime.
(10^3-1)*10^5+1=99900001 twin prime so a(5)=5.

Cf. A213883, A213884.

isA213882 := proc(b)
     local c,d,p;
    for c from 1 to 2*b-1 do
        for d from 0 to 9 do
            p := (10^c-d)*10^b-1 ;
            if isprime(p) and isprime(p+2) then
                return true;
            end if;
        end do:
    end do:
    return false ;
end proc:
for n from 1 to 2000 do
    if isA213882(n) then
        printf("%d,\n",n);
    end if;
end do; # R. J. Mathar, Jul 21 2012

cp's OEIS Frontend

A213884 For the smallest k >= 1, the smallest single-digit j such that (10^k-j)*10^n-1 is prime.

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A228144 Smallest k > n such that j10^k + m10^n - 1 is a prime number for at least a pair {j,m} with 0 < j < 10 and 0 < m < 10.

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A213882 Numbers b such that there is at least one number c and one single-digit number d such that (10^c-d)10^b-1 and (10^c-d)10^b+1 are twin primes with 0 < c < 2*b.

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Maple

cp's OEIS Frontend

A213884 For the smallest k >= 1, the smallest single-digit j such that (10^k-j)*10^n-1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A228144 Smallest k > n such that j*10^k + m*10^n - 1 is a prime number for at least a pair {j,m} with 0 < j < 10 and 0 < m < 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A213882 Numbers b such that there is at least one number c and one single-digit number d such that (10^c-d)*10^b-1 and (10^c-d)*10^b+1 are twin primes with 0 < c < 2*b.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A228144 Smallest k > n such that j10^k + m10^n - 1 is a prime number for at least a pair {j,m} with 0 < j < 10 and 0 < m < 10.

A213882 Numbers b such that there is at least one number c and one single-digit number d such that (10^c-d)10^b-1 and (10^c-d)10^b+1 are twin primes with 0 < c < 2*b.