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Smallest k > 1 such that prime(n)*k^prime(n)-1 is prime.
4

%I #19 Apr 25 2016 11:50:03

%S 2,2,4,8,10,56,46,6,4,102,98,90,52,12,28,418,426,482,38,28,140,26,354,

%T 882,756,268,146,4,260,76,48,288,1584,38,1102,2688,464,3500,16,5146,

%U 2562,2072,1020,726,306,1796,38,866,508,800,3480,132,750,4170

%N Smallest k > 1 such that prime(n)*k^prime(n)-1 is prime.

%C Start:

%C For primes, p < 25,000, for which p*k^p-1 is a prime:

%C k=1: just 3;

%C k=2: 2, 3, 751, 12379, …, ; indices: 1, 2, 133, 1478, …, ;

%C k=4: 2, 3, 5, 23, 107, 1973, 20747, …, ; indices: 1, 2, 3, 9, 28, 298, 2336, …, ;

%C k=6: 2, 3, 19, 107, 1999, …, ; indices: 1, 2, 8, 28, 303, …, ;

%C k=8: 2, 7, …, ; indices: 1, 4, …, ;

%C k=10: 2, 3, 11, 2843, …, ; indices: 1, 2, 5, 413, …, ; etc.

%C End. - _Robert G. Wilson v_, Aug 02 2014

%H Pierre CAMI, <a href="/A245597/b245597.txt">Table of n, a(n) for n = 1..300</a>

%e 2*2^2-1=7 prime so a(1)=2.

%e 3*2^3-1=23 prime so a(2)=2.

%e 5*2^5-1=159 composite.

%e 5*4^5-1=5119 prime so a(3)=4.

%t f[n_] := Block[{k = 2, p = Prime@ n}, While[ !PrimeQ[p*k^p - 1], k += 2]; k]; Array[f, 60] (* _Robert G. Wilson v_, Aug 02 2014 *)

%o (PFGW & SCRIPT)

%o SCRIPT

%o DIM i,0

%o DIM j

%o DIM n,-1

%o DIMS t

%o OPENFILEOUT myf,a(n).txt

%o LABEL loop1

%o SET i,i+1

%o IF i>300 THEN END

%o SET j,p(i)

%o SET n,0

%o LABEL loop2

%o SET n,n+2

%o SETS t,%d,%d,%d\,;i;j;n

%o PRP j*n^j-1,t

%o IF ISPRP THEN GOTO a

%o GOTO loop2

%o LABEL a

%o WRITE myf,t

%o GOTO loop1

%o (PARI) a(n) = k=2; while(!isprime(prime(n)*k^prime(n)-1), k+=2); k

%o vector(20, n, a(n)) \\ _Colin Barker_, Jul 27 2014

%Y Cf. A245598.

%K nonn

%O 1,1

%A _Pierre CAMI_, Jul 27 2014

%E Definition corrected by _Colin Barker_, Jul 27 2014