A111231 Numbers which are perfect powers m^k equal to the sum of m distinct primes.

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

Offset

1,2

Comments

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

Is a(n) = A076467(n) for all n > 1? - R. J. Mathar, May 22 2009

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

Links

Table of n, a(n) for n=1..45.

Example

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.

Prog

(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

Crossrefs

Cf. A007504, A076467.

Sequence in context: A377847 A036966 A076467 * A111307 A246549 A076405

Adjacent sequences: A111228 A111229 A111230 * A111232 A111233 A111234

Keyword

nonn

Author

Giovanni Teofilatto, Oct 28 2005

Extensions

Offset corrected by R. J. Mathar, May 25 2009

Edited by M. F. Hasler, May 25 2018

Status

approved