A225103 -id:A225103 - cp's OEIS frontend

A225104 Numbers that can be represented as a sum of two distinct nontrivial prime powers in three or more ways.

370, 650, 2210, 3770, 5330, 6290, 7202, 10370, 10730, 11570, 12410, 12818, 13130, 14690, 15170, 15650, 16250, 16490, 18122, 18530, 19370, 19610, 21170, 22490, 24050, 24650, 25010, 26690, 28730, 29930, 30290, 30770, 31610, 32810, 33410, 34970, 36482, 36490

Offset: 1

Alex Ratushnyak, Apr 28 2013

nonn

Indices of terms bigger than 2 in A225099.
Nontrivial prime powers are numbers of the form p^k where p is a prime number and k >= 2. That is, A025475 except the first term A025475(1) = 1.
It appears that all terms less than 2^34 are even.

Alois P. Heinz, Table of n, a(n) for n = 1..600

Cf. A025475, A225099, A225102, A225103.

isA025475not1 := proc(n)
    if n <= 1 then
        false;
    elif isprime(n) then
        false;
    elif nops(numtheory[factorset](n)) = 1 then
        true;
    else
        false;
    end if;
end proc:
A025475not1 := proc(n)
    option remember;
    local a;
    if n = 1 then
        4;
    else
        for a from procname(n-1)+1 do
            if isA025475not1(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
A225104w := proc(n)
    local a,i,ppi,ppj ;
    a := 0 ;
    for i from 1 do
        ppi := A025475not1(i) ;
        if ppi >= n/2 then
            break;
        end if;
        ppj := n-ppi ;
        if isA025475not1(ppj) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
for n from 1 do
    if A225104w(n) >= 3 then
        print(n) ;
    end if;
end do: # R. J. Mathar, Jun 13 2013

nn = 36490; p = Sort[Flatten[Table[Prime[n]^i, {n, PrimePi[Sqrt[nn]]}, {i, 2, Log[Prime[n], nn]}]]]; Transpose[Sort[Select[Tally[Flatten[Table[p[[i]] + p[[j]], {i, Length[p] - 1}, {j, i + 1, Length[p]}]]], #[[1]] <= nn && #[[2]] > 2 &]]][[1]] (* T. D. Noe, Apr 29 2013 *)

A282633 Numbers n such that n^2 + 1 is the sum of two proper prime powers (A246547) in more than one way.

Original entry on oeis.org

47, 73, 83, 133, 157, 173, 187, 191, 203, 217, 317, 319, 353, 437, 463, 467, 487, 499, 557, 577, 583, 593, 599, 613, 623, 697, 703, 727, 733, 767, 829, 857, 863, 871, 931, 983, 1013, 1027, 1033, 1067, 1087, 1097, 1123, 1139, 1177, 1267, 1279, 1321, 1327, 1333, 1363, 1403, 1409, 1433, 1453, 1477, 1487, 1493, 1507, 1517, 1543, 1567, 1603, 1607, 1613

Offset: 1

Robert Israel and Altug Alkan, Feb 19 2017

nonn

			83 is a term because 83^2 + 1 = 7^4 + 67^2 = 43^2 + 71^2.

Robert Israel, Table of n, a(n) for n = 1..2672

Cf. A002522, A225103, A246547.

N:= 10^8: # to get all terms <= sqrt(N-1).
PP:= sort([seq(seq(p^k, k=2..floor(log[p](N))), p = select(isprime, [2, seq(i, i=3..floor(sqrt(N)), 2)]))]):
npp:= nops(PP):
res:= {}: R:= 'R':
for i from 2 to npp do
   for j from 1 to i-1 do
     q:= PP[i]+PP[j];
     if q > N then break fi;
      if issqr(q-1) then
       if assigned(R[q]) then res:= res union {q}
        else R[q]:= 1
      fi fi
od od:
sort(convert(map(t -> sqrt(t-1), res),list));

A290135 Numbers that are the sum of two proper prime powers (A246547).

Original entry on oeis.org

8, 12, 13, 16, 17, 18, 20, 24, 25, 29, 31, 32, 33, 34, 35, 36, 40, 41, 43, 48, 50, 52, 53, 54, 57, 58, 59, 64, 65, 68, 72, 73, 74, 76, 80, 81, 85, 89, 90, 91, 96, 97, 98, 106, 108, 113, 125, 128, 129, 130, 132, 133, 134, 136, 137, 141, 144, 145, 146, 148, 150, 152, 153, 155, 157, 160, 162, 170, 173, 174, 177, 178

Offset: 1

Ilya Gutkovskiy, Jul 20 2017

nonn

Is 2213 the largest prime term that can be expressed as the sum of two proper prime powers in more than one way? - Altug Alkan, Jul 22 2017

			13 is in the sequence because 13 = 2^2 + 3^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014091, A070049, A071330, A071331, A225102, A225103, A246547.

N:= 1000: # to get all terms <= N
P:= select(isprime, [$2..floor(sqrt(N))]):
PP:= {seq(seq(p^j, j=2..floor(log[p](N))),p=P)}:
A:= select(`<=`,{seq(seq(PP[i]+PP[j],j=1..i),i=1..nops(PP))},N):
sort(convert(A,list)); # Robert Israel, Jul 21 2017

nmax = 180; f[x_] := Sum[Boole[PrimePowerQ[k] && PrimeOmega[k] > 1] x^k, {k, 1, nmax}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

Exponents in expansion of (Sum_{k>=1} x^A246547(k))^2.

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

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Feb 17 2017

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

			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.

Cf. A225099, A225102, A225103, A246547.

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

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

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

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 *)

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

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

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A225104 Numbers that can be represented as a sum of two distinct nontrivial prime powers in three or more ways.

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A282633 Numbers n such that n^2 + 1 is the sum of two proper prime powers (A246547) in more than one way.

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A290135 Numbers that are the sum of two proper prime powers (A246547).

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

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