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A068704
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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.
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3
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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
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
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)
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MAPLE
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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
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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base,hard,more,nonn
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AUTHOR
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EXTENSIONS
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Entry revised by N. J. A. Sloane, May 22 2014 to reflect the fact that a(11) is presently unknown.
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STATUS
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approved
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