|
%I #30 Jun 25 2021 23:37:58
%S 0,0,0,0,0,1,1,0,1,1,3,1,1,1,2,1,1,1,2,1,3,1,2,0,1,1,4,2,2,2,1,3,2,2,
%T 3,1,2,4,4,2,2,1,2,2,4,2,3,1,3,2,4,2,2,2,3,4,2,1,2,1,2,3,3,1,2,3,3,4,
%U 4,2,2,2,2,1,5,1,4,2,3,3,2,1,5,2,1,4,4,3,2,1,2,4,3,2,3,2,2,4,2,2,2,2,3,2,6,2,4,2,2,4,5,2,3,1,3,3,5,2,3,1,2,4,4,3,3,2,1,6
%N Number of ways to write n as a sum of a perfect power (>1) and a prime.
%C In this sequence, "perfect power" does not include 0 or 1, "prime" does not include 1. Both "perfect power" and "prime" must be positive.
%C 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.
%C 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.
%C This conjecture is not true: a(1771561) = 0. (See A119748)
%C Another conjecture: For every number m>=0, there is a number k such that a(n)>=m for all n>=k.
%C Another conjecture: Except for k=2, first occurrence of k must be earlier then first occurrence of k+1.
%C For n such that a(n) = 0, see A119748.
%C For n such that a(n) = 1, see the following a-file of this sequence.
%H Eric Chen and Charles R Greathouse IV, <a href="/A253238/b253238.txt">Table of n, a(n) for n = 1..10000</a> (first 3000 terms from Chen)
%H Eric Chen, <a href="/A253238/a253238.txt">The unique way to write n as a sum of a perfect power (>1) and a prime for those n such a(n)=1</a>
%t 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
%o (PARI) a(n) = sum(k=1, n-1, ispower(k) && isprime(n-k))
%o (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
%Y Cf. A196228, A119748, A109925, A118954, A064272, A064233, A002471, A014090.
%K nonn,easy
%O 1,11
%A _Eric Chen_, May 17 2015
|
