OFFSET

1,11

COMMENTS

In this sequence, "perfect power" does not include 0 or 1, "prime" does not include 1. Both "perfect power" and "prime" must be positive.

In the past, I conjectured that a(n) > 0 for all n>24, but this is not true. My PARI program found that a(1549) = 0.

I also asked which a(n) are 1. For example, 331 is a de Polignac number (A006285), so it cannot be written as 2^n+p with p prime, and 331-6^n must divisible by 5, 331-10^n must divisible by 3, ..., 331-18^2 = 331-324 = 7 is prime (and it is the only prime of the form 331-m^n, with m, n natural numbers, m>1, n>1), so a(331) = 1. Similarly, a(3319) = 1. Conjecture: a(n) > 1 for all n > 3319.

This conjecture is not true: a(1771561) = 0. (See A119748)

Another conjecture: For every number m>=0, there is a number k such that a(n)>=m for all n>=k.

Another conjecture: Except for k=2, first occurrence of k must be earlier then first occurrence of k+1.

For n such that a(n) = 0, see A119748.

For n such that a(n) = 1, see the following a-file of this sequence.

LINKS

Eric Chen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 3000 terms from Chen)

MATHEMATICA

nn = 128; pwrs = Flatten[Table[Range[2, Floor[nn^(1/ex)]]^ex, {ex, 2, Floor[Log[2, nn]]}]]; pp = Prime[Range[PrimePi[nn]]]; t = Table[0, {nn}]; Do[ t[[i[[1]]]] = i[[2]], {i, Tally[Sort[Select[Flatten[Outer[Plus, pwrs, pp]], # <= nn &]]]}]; t

PROG

(PARI) a(n) = sum(k=1, n-1, ispower(k) && isprime(n-k))

(PARI) a(n)=sum(e=2, log(n)\log(2), sum(b=2, sqrtnint(n, e), isprime(n-b^e)&&!ispower(b))) \\ Charles R Greathouse IV, May 28 2015

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Eric Chen, May 17 2015

STATUS

approved