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Smallest primes such that every partial sum is an n-th power.
2

%I #5 Jan 06 2024 21:57:40

%S 2,7,503,1889,30367

%N Smallest primes such that every partial sum is an n-th power.

%C The sequence is complete. Proof: We must find a prime p and an integer x such that p = x^6 - (2+7+503+1889+30367) = x^6 - 32768 = (x^2-32)*(x^4+32*x^2+1024). Since p is prime, we must have p=1*p, therefore we can only have x=sqrt(33) to make p = (1)*(3169). However, sqrt(33) is not an integer. Therefore we can conclude that there is no prime p satisfying the equation. - Francois Jooste (pin(AT)myway.com), Mar 09 2003

%F a(n) = x^n - Sum_{i=1..n-1} a(i), for some integer x and a(n) prime for all n. - Francois Jooste (pin(AT)myway.com), Mar 09 2003

%e a(3) = 503, 2+7+503 = 512 = 8^3.

%K fini,full,nonn

%O 1,1

%A _Amarnath Murthy_, Aug 15 2002