A257460 -id:A257460

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A257459

Let b_k=1...1 consist of k>0 1's. Then a(n) is the smallest k such that the concatenation prime(n)b_k is prime, or a(n)=0 if there is no such prime.

+10
3

2, 1, 5, 1, 17, 1, 8, 1, 2, 6, 1, 0, 2, 1, 3, 9, 18, 4, 210, 6, 7, 3, 2, 6, 1, 2, 1, 2, 1, 2, 4, 3, 2, 24, 3, 1, 1, 6, 5, 11, 2, 1, 11, 1, 12, 6, 1, 7, 3, 39, 2, 2, 1, 2, 9, 3, 5, 1, 6, 2, 3, 2, 180, 3, 15, 17, 24, 1, 5, 1, 2, 2, 1, 64, 7, 6, 3, 24, 2, 1, 2, 1, 6, 16, 1, 9, 8, 6, 17, 4, 6, 2, 1, 9, 30, 2, 6, 44, 1, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The only unknown terms less than 10000, tested to 15000, are for n: 284, 714, 1257, 1618, 2248, 2450, 2779, 3886, 3891, 4007, 4359, 4784, 4912, 5364, 6108, 6356, 6371, 7570, 7668, 8446, 9606.

Prime(12)=37 and b_k for k == 2 (mod 3), the concatenation is divisible by 3; for k == 1 (mod 3), the concatenation is divisible by either 7 or 13; and finally for k == 0 (mod 3), the concatenation is divisible by 37.

LINKS

Table of n, a(n) for n=1..100.

Vladimir Shevelev and Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found.

FORMULA

a(n)=k for the least k such that p(n)*10^k+(10^k-1)/9 is prime, where p(n) is the n_th prime.

MATHEMATICA

f[n_] := Block[{k = 1, p = Prime[n]}, While[ !PrimeQ[p*10^k + (10^k - 1)/9], k++]; k]; f[12] = 0; Array[f, 100]

CROSSREFS

Cf. A232210, A257460, A257461.

KEYWORD

nonn,base

AUTHOR

Vladimir Shevelev and Robert G. Wilson v, Apr 24 2015

STATUS

approved

A257461

Let b_k=9...9 consist of k>0 9's. Then a(n) is the smallest k such that the concatenation prime(n)b_k is prime, or a(n)=0 if there is no such prime.

+10
3

1, 0, 1, 1, 5, 1, 1, 1, 1, 2, 28, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 3, 1, 2, 90, 1, 1, 2, 8, 2, 1, 1, 2, 1, 1, 2, 1, 4, 6, 8, 3, 2, 3, 4, 1, 1, 5, 1, 5, 60, 1, 1, 5, 6, 1, 2, 1, 1, 2, 1, 10, 1, 1, 1, 5, 2, 1, 3, 4, 1, 1, 2, 4, 1, 3, 4, 3, 2, 1, 1, 2, 1, 6, 1, 5, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

The only unknown terms less than 10000, tested to 25000, are for n: 87, 5744, 8041, 9533.

For p(87)=449, the concatenation is divisible by 11 if k is odd and is divisible by 7 if k == 4 (mod 6).

LINKS

Table of n, a(n) for n=1..86.

Vladimir Shevelev and Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found.

FORMULA

a(n)=k for the least k such that p(n)*10^k+10^k-1 is prime, where p(n) is the n_th prime.

MATHEMATICA

f[n_] := Block[{k = 1, p = Prime[n]}, While[ !PrimeQ[p*10^k + 10^k - 1], k++]; k]; f[2] = 0; Array[f, 86]

CROSSREFS

Cf. A257459, A232210, A257460.