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A282533 Primes that are the sum of two proper prime powers (A246547) in more than one way. 0
41, 89, 113, 137, 593, 857, 2213 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 26 16:44 EDT 2021. Contains 346294 sequences. (Running on oeis4.)