%I #41 May 23 2014 12:15:43
%S 11,421,31,41,2551,61,71,6481,8191,101
%N 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.
%C Conjecture: a(n) > 0, that is, for every n there exists a k which yields a prime.
%C 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
%e a(3) = 31 as the concatenation of 3^1 and 1 is prime, so here k = 1.
%e a(5) = 2551= concatenation of 5^2, 5 and 1 is prime, so here k = 2.
%e The known values are as follows:
%e n k a(n)
%e 1 1 11
%e 2 2 421
%e 3 1 31
%e 4 1 41
%e 5 2 2551
%e 6 1 61
%e 7 1 71
%e 8 2 6481
%e 9 2 8191
%e 10 1 101
%e 11 ? ? (see A242645)
%e 12 4 207361728144121
%e 13 1 131
%e 14 ? ?
%e 15 1 151
%e 16 4 655364096256161
%e 17 2 289171
%e 18 1 181
%e 19 1 191
%e 20 ? ?
%e 21 1 211
%e 22 ? ?
%e 23 ? ?
%e 24 1 241
%e 25 1 251
%e 26 ? ?
%e 27 1 271
%e 28 1 281
%e 29 4 70728124389841291 (_Jayanta Basu_, May 21 2013)
%e 30 6 7290000002430000081000027000900301 (_Jayanta Basu_, May 21 2013)
%e 31 1 311
%e 32 2 1024321
%e 33 1 331
%e 34 33 a(n) is an 877-digit number (_Ray Chandler_, Apr 06 2014)
%e 35 6 1838265625525218751500625428751225351 (_Jayanta Basu_, May 21 2013)
%p for i from 1 to 23 do k := 0:
%p c := true:
%p 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:
%p k := k+1: if(a>10^500) then a := -k; break: end if:
%p if(isprime(a)) then c := false: end if: end do:
%p 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
%t 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]
%Y Cf. A242645 (for a(11)).
%K base,hard,more,nonn
%O 1,1
%A _Amarnath Murthy_, Mar 04 2002
%E Corrected by _Lior Manor_, May 07 2006
%E Entry revised by _N. J. A. Sloane_, May 22 2014 to reflect the fact that a(11) is presently unknown.