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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.

%I A068704 #41 May 23 2014 12:15:43
%S A068704 11,421,31,41,2551,61,71,6481,8191,101
%N 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.
%C A068704 Conjecture: a(n) > 0, that is, for every n there exists a k which yields a prime.
%C A068704 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 A068704 a(3) = 31 as the concatenation of 3^1 and 1 is prime, so here k = 1.
%e A068704 a(5) = 2551= concatenation of 5^2, 5 and 1 is prime, so here k = 2.
%e A068704 The known values are as follows:
%e A068704 n   k    a(n)
%e A068704 1   1    11
%e A068704 2   2    421
%e A068704 3   1    31
%e A068704 4   1    41
%e A068704 5   2    2551
%e A068704 6   1    61
%e A068704 7   1    71
%e A068704 8   2    6481
%e A068704 9   2    8191
%e A068704 10  1    101
%e A068704 11  ?    ? (see A242645)
%e A068704 12  4    207361728144121
%e A068704 13  1    131
%e A068704 14  ?    ?
%e A068704 15  1    151
%e A068704 16  4    655364096256161
%e A068704 17  2    289171
%e A068704 18  1    181
%e A068704 19  1    191
%e A068704 20  ?    ?
%e A068704 21  1    211
%e A068704 22  ?    ?
%e A068704 23  ?    ?
%e A068704 24  1    241
%e A068704 25  1    251
%e A068704 26  ?    ?
%e A068704 27  1    271
%e A068704 28  1    281
%e A068704 29  4    70728124389841291 (_Jayanta Basu_, May 21 2013)
%e A068704 30  6    7290000002430000081000027000900301 (_Jayanta Basu_, May 21 2013)
%e A068704 31  1    311
%e A068704 32  2    1024321
%e A068704 33  1    331
%e A068704 34  33   a(n) is an 877-digit number (_Ray Chandler_, Apr 06 2014)
%e A068704 35  6    1838265625525218751500625428751225351 (_Jayanta Basu_, May 21 2013)
%p A068704 for i from 1 to 23 do k := 0:
%p A068704 c := true:
%p A068704 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 A068704 k := k+1: if(a>10^500) then a := -k; break: end if:
%p A068704 if(isprime(a)) then c := false: end if: end do:
%p A068704 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 A068704 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 A068704 Cf. A242645 (for a(11)).
%K A068704 base,hard,more,nonn
%O A068704 1,1
%A A068704 _Amarnath Murthy_, Mar 04 2002
%E A068704 Corrected by _Lior Manor_, May 07 2006
%E A068704 Entry revised by _N. J. A. Sloane_, May 22 2014 to reflect the fact that a(11) is presently unknown.