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A111231
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Numbers which are perfect powers m^k equal to the sum of m distinct primes.
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5
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0, 8, 16, 27, 32, 64, 81, 125, 128, 216, 243, 256, 343, 512, 625, 729, 1000, 1024, 1296, 1331, 1728, 2048, 2187, 2197, 2401, 2744, 3125, 3375, 4096, 4913, 5832, 6561, 6859, 7776, 8000, 8192, 9261, 10000, 10648, 12167, 13824, 14641, 15625, 16384, 16807
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OFFSET
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1,2
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COMMENTS
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Perfect powers m^k with k >= 3, m = 0 or m > 1.
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
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LINKS
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EXAMPLE
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a(1) = 0 because 0 = 0^2 = 0^3 is the sum of 0 primes;
a(2) = 8 because 8 = 2^3 = 3 + 5, sum of 2 primes;
a(3) = 16 because 16 = 2^4 = 3 + 13, sum of 2 primes.
a(4) = 27 because 27 = 3^3 = 3 + 11 + 13, sum of 3 primes.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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