1,1

For every n, a(n) must be equal to 1 or 2 (mod 4) because Sum[k^k,{k,a(n)}] must be odd. Any additional terms are greater than 5368 with the next prime having more than 20025 digits. - Farideh Firoozbakht (f.firoozbakht(AT)sci.ui.ac.ir), Aug 09 2003

Soundararajan finds an asymptotic upper bound of log k / log log k prime numbers of the form 1^1 + 2^2 + ... + n^n less than k; that is, n = O(log a(n) / log log a(n)). - Charles R Greathouse IV, Aug 27 2008

According to Andersen, the next term is larger than 28000, cf. link. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 01 2009]

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 308.

K. Soundararajan, "Primes in a Sparse Sequence", Journal of Number Theory 43:2 (1993), pp. 220-227.

C. Rivera, Prime puzzle #404. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Mar 01 2009]

Mitrinovic gives upper bound on density.

v={}; Do[If[(Mod[n, 4]==1||Mod[n, 4]==2)&&PrimeQ[Sum[k^k, {k, n}]], v=Insert[v, n, -1]; Print[v]], {n, 5368}]

(PARI) s=0; for(k=1, 1320, s=s+k^k; if(isprime(s), print1(k, ", ")))

Cf. A073826 (corresponding primes), A001923 (Sum k^k, k=1..n).

Sequence in context: A057250 A056643 A057256 this_sequence A015891 A160645 A026344

Adjacent sequences: A073822 A073823 A073824 this_sequence A073826 A073827 A073828

nonn

Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 13 2002

Any additional terms are greater than 1320 with the next prime having more than 4120 digits.

No terms out to 3000. The next term would yield a prime with over 10000 digits. - John Sillcox (johnsillcox(AT)hotmail.com), Aug 05 2003