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Numbers which are perfect powers m^k equal to the sum of m distinct primes.
5

%I #21 Jun 01 2018 01:50:52

%S 0,8,16,27,32,64,81,125,128,216,243,256,343,512,625,729,1000,1024,

%T 1296,1331,1728,2048,2187,2197,2401,2744,3125,3375,4096,4913,5832,

%U 6561,6859,7776,8000,8192,9261,10000,10648,12167,13824,14641,15625,16384,16807

%N Numbers which are perfect powers m^k equal to the sum of m distinct primes.

%C Perfect powers m^k with k >= 3, m = 0 or m > 1.

%C Is a(n) = A076467(n) for all n > 1? - _R. J. Mathar_, May 22 2009

%C A sum of m distinct primes is >= A007504(m) ~ m^2(log m)/2 > m^2, also for small m, therefore the second condition excludes squares m^2. On the other hand, considering results related to Goldbach's conjecture (e.g., every even number >= 4 is the sum of at most 4 primes), it is increasingly improbable that some m^k with k >= 3 is not the sum of m primes. This explains the first comment - but can it be rigorously proved? - _M. F. Hasler_, May 25 2018

%e a(1) = 0 because 0 = 0^2 = 0^3 is the sum of 0 primes;

%e a(2) = 8 because 8 = 2^3 = 3 + 5, sum of 2 primes;

%e a(3) = 16 because 16 = 2^4 = 3 + 13, sum of 2 primes.

%e a(4) = 27 because 27 = 3^3 = 3 + 11 + 13, sum of 3 primes.

%o (PARI) is(n,d)={if(d=ispower(n), fordiv(d,e,e>1&&forvec(v=vector(d=sqrtnint(n,e)-1,i,[1,primepi((n-1)\2-d+3)]),prime(v[#v])<(d=n-vecsum(apply(i->prime(i),v)))&&isprime(d)&&return(1),2)), !n)} \\ _M. F. Hasler_, May 25 2018

%Y Cf. A007504, A076467.

%K nonn

%O 1,2

%A _Giovanni Teofilatto_, Oct 28 2005

%E Offset corrected by _R. J. Mathar_, May 25 2009

%E Edited by _M. F. Hasler_, May 25 2018