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Smallest k such that 1 + Sum{j=1..n} k^(2*j-1) is prime.
3

%I #11 Jul 23 2024 14:51:55

%S 1,1,2,1,2,1,10,2,2,1,2,1,48,182,2,1,60,1,10,42,2,1,102,12,4,12,110,1,

%T 12,1,100,5,28,18,144,1,102,9,2,1,30,1,186,110,130,1,566,23,1234,2,12,

%U 1,336,103,142,341,1104,1,444,1,22,119,2,45,14,1,84,23,238,1,936,1,78,12

%N Smallest k such that 1 + Sum{j=1..n} k^(2*j-1) is prime.

%C a(n) = 1 if and only if n is in A006093 (primes minus 1), so 1 occurs infinitely often.

%e Consider n = 7. 1 + Sum{j=1...7} k^(2*j-1) evaluates to 8, 10923, 1793614, 71582789, 1271565756, 13433856703, 98907531458, 558482096649, 2573639151184, 10101010101011 for k = 1, ..., 10. Only the last of these numbers, 1+10+10^3+10^5+10^7+10^9+10^11+10^13 = 10101010101011, is prime, hence a(7) = 10.

%t f[n_] := Block[{k = 1}, While[ !PrimeQ[Sum[k^(2j - 1), {j, n}] + 1], k++ ]; k]; Array[f, 74] (* _Robert G. Wilson v_, Dec 17 2006 *)

%o (PARI) a(n)={my(k=1);while(!isprime(1+sum(j=1,n,k^(2*j-1))),k++); k} \\ _Klaus Brockhaus_, Dec 16 2006

%Y Cf. A006093, A124205-A124209, A124164, A124178, A124181, A124185-A124187, A124189, A124200, A124154, A124163.

%K nonn

%O 1,3

%A _Artur Jasinski_, Dec 13 2006, Dec 14 2006

%E Edited and extended by _Klaus Brockhaus_, Dec 16 2006