A282533 Primes that are the sum of two proper prime powers (A246547) in more than one way.

41, 89, 113, 137, 593, 857, 2213

Offset

1,1

Comments

Primes of the form 2^k + p^e in more than one way where p is an odd prime (e > 1, k > 1).

Prime terms in A225103.

29 = 2^4 + 5^2 = 2 + 3^3 is a border case not included in this sequence - Olivier Gérard, Feb 25 2019

a(8) > 10^8 if it exists. - Robert Israel, Feb 17 2017

a(8) > 10^18 if it exists. - Charles R Greathouse IV, Feb 19 2017

Links

Table of n, a(n) for n=1..7.

Example

41 = 2^4 + 5^2 = 2^5 + 3^2.
89 = 2^3 + 3^4 = 2^6 + 5^2.
113 = 2^5 + 3^4 = 2^6 + 7^2.
137 = 2^7 + 3^2 = 2^4 + 11^2.
593 = 2^9 + 3^4 = 2^6 + 23^2.
857 = 2^7 + 3^6 = 2^4 + 29^2.
2213 = 2^4 + 13^3 = 2^2 + 47^2.

Maple

N:= 10^6: # to get all terms <= N
B:= Vector(N):
C:= Vector(N):
for k from 2 to ilog2(N) do B[2^k]:= 1 od:
p:= 2:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  for k from 2 to floor(log[p](N)) do C[p^k]:= 1 od:
od:
R:= SignalProcessing:-Convolution(B, C):
select(t -> isprime(t) and R[t-1] > 1.5, [seq(i, i=3..N, 2)]); # Robert Israel, Feb 17 2017

Mathematica

Select[Prime@ Range[10^3], Function[n, Count[Transpose@{n - #, #}, w_ /; Times @@ Boole@ Map[And[PrimePowerQ@ #, ! PrimeQ@ #] &, w] > 0] >= 2 &@ Range[4, Floor[n/2]]]] (* or *)
With[{n = 10^8}, Keys@ Select[#, Length@ # > 1 &] &@ GroupBy[#, First] &@ SortBy[Transpose@ {Map[Total, #], #}, First] &@ Select[Union@ Map[Sort, Tuples[#, 2]], PrimeQ@ Total@ # &] &@ Flatten@ Map[#^Range[2, Log[#, Prime@ n]] &, Array[Prime@ # &, Floor@ Sqrt@ n]]] (* Michael De Vlieger, Feb 19 2017, latter program Version 10 *)

Prog

(MATLAB)
N = 10^8; % to get all terms <= N
C = sparse(1, N);
for p = primes(sqrt(N))
  C(p .^ [2:floor(log(N)/log(p))]) = 1;
end
R = zeros(1, N);
for k = 2: floor(log2(N))
  R((2^k+1):N) = R((2^k+1):N) + C(1:(N-2^k));
end
P = primes(N);
P(R(P) > 1.5) % Robert Israel, Feb 17 2017
(PARI) is(n) = if(!ispseudoprime(n), return(0), my(x=n-1, y=1, i=0); while(y < x, if(isprimepower(x) > 1 && isprimepower(y) > 1, if(i==0, i++, return(1))); y++; x--)); 0 \\ Felix Fröhlich, Feb 18 2017
(PARI) has(p)=my(t, q); p>40 && sum(k=2, logint(p-9, 2), t=2^k; sum(e=2, logint(p-t, 3), ispower(p-t, e, &q) && isprime(q)))>1
list(lim)=my(v=List(), t, q); lim\=1; if(lim<9, lim=9); for(k=2, logint(lim-9, 2), t=2^k; for(e=2, logint(lim-t, 3), forprime(p=3, sqrtnint(lim-t, e), q=t+p^e; if(isprime(q) && has(q), listput(v, q))))); Set(v) \\ Charles R Greathouse IV, Feb 18 2017

Crossrefs

Cf. A225099, A225102, A225103, A246547.

Cf. A115231 (prime numbers which cannot be written as 2^a + p^b, b>=0)

Sequence in context: A276302 A087939 A242708 * A199983 A241081 A188173

Adjacent sequences: A282530 A282531 A282532 * A282534 A282535 A282536

Keyword

nonn,more

Author

Altug Alkan, Feb 17 2017

Status

approved