OFFSET
1,1
COMMENTS
Except for the terms 2, 5, 13, 29, 139, the exponent k satisfies k >= 4. More generally, if Q(j) = j^k + (j-1)^k + ... + 2^k is a term, then j-1 is a divisor of A064538(k). This is because (j-1) is a factor of Q(j) and thus Q(j) is prime only if j-1 is a divisor of the denominator of Q(j), i.e. A064538(k). Thus for each k there is only a finite number of values of j to check. This provides an efficient algorithm to find terms of this sequence by looking only for primes in the numbers H_{j,-k) - 1 = j^k + (j-1)^k + ... + 2^k for j-1 a divisor of A064538(k). - Chai Wah Wu, Mar 06 2018
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..45 (all terms < 10^1000).
EXAMPLE
2 = 2^1;
5 = 3^1 + 2^1;
13 = 3^2 + 2^2;
29 = 4^2 + 3^2 + 2^2;
97 = 3^4 + 2^4;
139 = 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2;
353 = 4^4 + 3^4 + 2^4;
4889 = 4^6 + 3^6 + 2^6;
72353 = 4^8 + 3^8 + 2^8;
MATHEMATICA
With[{nn = 350}, Sort@ Flatten@ Map[Select[#, PrimeQ] &, Table[Total[Range[j, 1, -1]^k] - 1, {j, 2, nn}, {k, nn - j}]]] (* Michael De Vlieger, Feb 03 2018 *)
PROG
(PARI) limit=100000; v=vector(limit); for(n=1, ceil((-1+(1+8*limit)^(1/2))/2), for(k=1, logint(limit, n+0^(n-1)), a=sum(i=1, n, i^k)-1; if(isprime(a)&&a<limit+1, v[a]=1, ))); for(a=1, limit, if(v[a], print1(a", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gionata Neri, Feb 03 2018
EXTENSIONS
a(10)-a(15) from Michael De Vlieger, Feb 03 2018
a(16)-a(17) from Chai Wah Wu, Mar 07 2018
STATUS
approved