A068704 a(n) = smallest prime obtained as the concatenation of n^k, n^(k-1), n^(k-2), ..., n^2, n, 1 for some k >= 1; or 0 if no such prime exists.

11, 421, 31, 41, 2551, 61, 71, 6481, 8191, 101

Offset

1,1

Comments

Conjecture: a(n) > 0, that is, for every n there exists a k which yields a prime.

If we stop the search when the concatenation reaches 10^500 then the sequence would continue: ?, 207361728144121, 131, ?, 151, 655364096256161, 289171, 181, 191, ?, 211, ?, ?. - Sascha Kurz, Mar 27 2002

Links

Table of n, a(n) for n=1..10.

Example

a(3) = 31 as the concatenation of 3^1 and 1 is prime, so here k = 1.
a(5) = 2551= concatenation of 5^2, 5 and 1 is prime, so here k = 2.
The known values are as follows:
n   k    a(n)
1   1    11
2   2    421
3   1    31
4   1    41
5   2    2551
6   1    61
7   1    71
8   2    6481
9   2    8191
10  1    101
11  ?    ? (see A242645)
12  4    207361728144121
13  1    131
14  ?    ?
15  1    151
16  4    655364096256161
17  2    289171
18  1    181
19  1    191
20  ?    ?
21  1    211
22  ?    ?
23  ?    ?
24  1    241
25  1    251
26  ?    ?
27  1    271
28  1    281
29  4    70728124389841291 (Jayanta Basu, May 21 2013)
30  6    7290000002430000081000027000900301 (Jayanta Basu, May 21 2013)
31  1    311
32  2    1024321
33  1    331
34  33   a(n) is an 877-digit number (Ray Chandler, Apr 06 2014)
35  6    1838265625525218751500625428751225351 (Jayanta Basu, May 21 2013)

Maple

for i from 1 to 23 do k := 0:
c := true:
while(c) do a := i^k:for j from k-1 to 0 by -1 do a := i^j+10^(floor(log(i^j)/log(10)+0.000001)+1)*a; end do:
k := k+1: if(a>10^500) then a := -k; break: end if:
if(isprime(a)) then c := false: end if: end do:
b[i] := a: end do: seq(b[k], k=1..23); # Warning: program may not compute a(n). - N. J. A. Sloane, May 22 2014

Mathematica

Table[k=1; While[!PrimeQ[x=FromDigits[Flatten[IntegerDigits[Reverse[n^Range[0, k]]]]]] && k<30, k++]; If[k==30, x=0]; x, {n, 34}] (* Jayanta Basu, May 21 2013 *) [Warning: program may not compute a(n). - N. J. A. Sloane, May 22 2014]

Crossrefs

Cf. A242645 (for a(11)).

Sequence in context: A197599 A197983 A351611 * A090558 A068135 A197770

Adjacent sequences: A068701 A068702 A068703 * A068705 A068706 A068707

Keyword

base,hard,more,nonn

Author

Amarnath Murthy, Mar 04 2002

Extensions

Corrected by Lior Manor, May 07 2006

Entry revised by N. J. A. Sloane, May 22 2014 to reflect the fact that a(11) is presently unknown.

Status

approved