A225102 -id:A225102 - cp's OEIS frontend

A225103 Numbers that can be represented as a sum of two distinct nontrivial prime powers in two or more ways.

36, 41, 57, 89, 113, 129, 130, 137, 153, 177, 185, 297, 305, 368, 370, 377, 410, 425, 537, 545, 561, 593, 633, 650, 657, 850, 857, 868, 873, 890, 969, 986, 1010, 1130, 1385, 1490, 1690, 1730, 1802, 1865, 1881, 1898, 1970, 2210, 2213, 2217, 2236, 2330, 2337

Offset: 1

Alex Ratushnyak, Apr 28 2013

nonn

Indices of terms bigger than 1 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.
Only 267 of the terms below 2^34 are odd.

			36 = 32 + 4 = 27 + 9, so 36 is in the sequence.
41 = 32 + 9 = 25 + 16, so 41 is in the sequence.

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

Cf. A025475, A225099, A225102.

N:= 10^4: # to get all terms <= N
PP:= [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 assigned(R[q]) then res:= res union {q}
    else R[q]:= 1
    fi
od od:
sort(convert(res,list)); # Robert Israel, Feb 17 2017

nn = 2409; 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]] > 1 &]]][[1]] (* T. D. Noe, Apr 29 2013 *)

A282550 Perfect powers that are the sum of two distinct proper prime powers (A246547).

Original entry on oeis.org

25, 36, 81, 125, 144, 196, 324, 512, 576, 1089, 2304, 2744, 2916, 5041, 9216, 14884, 16641, 26244, 36864, 51984, 147456, 236196, 589824, 941192, 1196836, 2125764, 2359296, 9437184, 19131876, 37748736, 67125249, 150994944, 172186884, 322828856, 603979776

Offset: 1

Altug Alkan, Feb 18 2017

nonn

Intersection of A001597 and A225102. - Michel Marcus, Feb 18 2017

Terms t of A001597 such that A225099(t) > 0. - Felix Fröhlich, Feb 18 2017

			512 = 2^9 is a term because 2^9 = 7^3 + 13^2.

Giovanni Resta, Table of n, a(n) for n = 1..45 (terms < 2*10^11)

Cf. A001597, A225099, A225102, A225106, A246547.

Select[Union@ Map[Total, Subsets[With[{nn = 10^6}, Complement[ Select[ Range@ nn, PrimePowerQ], Prime[Range[PrimePi@ nn]]]], {2}]], # == 1 ||
GCD @@ FactorInteger[#][[All, 2]] > 1 &] (* Michael De Vlieger, Feb 18 2017, after Harvey P. Dale at A246547 *)

is(n) = if(!ispower(n), return(0), my(x=n-1, y=1); while(y < x, if(isprimepower(x) && isprimepower(y) && !ispseudoprime(x) && !ispseudoprime(y), return(1)); y++; x--)); 0 \\ Felix Fröhlich, Feb 18 2017

More terms from Felix Fröhlich, Feb 18 2017

a(28)-a(35) from Giovanni Resta, May 07 2017

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

Original entry on oeis.org

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 *)

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

A282578 Least k such that k^n is the sum of two distinct proper prime powers (A246547), or 0 if no such k exists.

Original entry on oeis.org

12, 5, 5, 3, 62

Offset: 1

Altug Alkan, Feb 20 2017

Corresponding values of k^n are 12, 25, 125, 81, 916132832, ...

			a(1) = 12 because 12 = 2^2 + 2^3.
a(2) = 5 because 5^2 = 2^4 + 3^2.
a(3) = 5 because 5^3 = 2^2 + 11^2.
a(4) = 3 because 3^4 = 2^5 + 7^2.
a(5) = 62 because 62^5 = 31^5 + 31^6.
a(9) = 2 because 2^9 = 7^3 + 13^2.

Wikipedia, Beal's conjecture

Cf. A001597, A225102, A246547, A282550.

from sympy import nextprime, perfect_power
def ppupto(limit): # distinct proper prime powers <= limit
    p = 2; p2 = pk = p*p; pklist = []
    while p2 <= limit:
        while pk <= limit: pklist.append(pk); pk *= p
        p = nextprime(p); p2 = pk = p*p
    return sorted(pklist)
def sum_of_pp(n):
    pp = ppupto(n); ppset = set(pp)
    for p in pp:
        if p > n//2: break
        if n - p in ppset and n - p != p: return True
    return False
def a(n):
    k = 2
    while not sum_of_pp(k**n): k += 1
    return k
print([a(n) for n in range(1, 6)]) # Michael S. Branicky, Dec 05 2021

a(p) <= 2 * (2^p - 1) where p is in A000043 since (2^p - 1)^p + (2^p - 1)^(p + 1) = (2 * (2^p - 1))^p.

A282686 Least sum of two proper prime powers (A246547) that is the product of n distinct primes.

Original entry on oeis.org

13, 33, 130, 966, 14322, 81510, 3530730, 117535110, 2211297270, 131031070170, 1295080356570, 163411918786830, 3389900689405230, 414524121952915590, 2951531806477464210, 754260388389042905370

Offset: 1

Altug Alkan, Feb 20 2017

Least value of A225102 that is the product of n distinct primes.
From Jon E. Schoenfield, Mar 18 2017: (Start)
For each n, we can write a(n) = p^j + q^k where p and q are prime and 2 <= j <= k; since a(n) is squarefree, p and q are distinct.
Suppose j and k are both even. Then a(n) cannot have any prime factor f such that f == 3 (mod 4) (see A002145). Thus, a(n) is the product of n distinct terms of {2, 5, 13, 17, 29, 37, 41, ...} = A002313, so a(n) >= Product_{i=1..n} A002313(i) = A185952(n).
In fact, however, a(n) < A185952(n) for n = 4..15, and it seems nearly certain that this holds for all n > 3. In any case, if we search for a(n) by generating products of n distinct primes and, for each such product P, testing whether there exists a solution for P = p^j + q^k, then we need not consider solutions in which both j and k are even unless P >= A185952(n).
Additionally, since the sum of any two cubes that is divisible by 3 is also divisible by 9 (hence nonsquarefree), any P that is divisible by 3 cannot be the sum of two cubes, so the exponents j and k cannot both be divisible by 3. (Every P < 2*5*7*11*...*prime(n+1) = A002110(n+1)/3 is divisible by 3.) Thus, for every P that is divisible by 3 and < A185292(n), we can rule out every ordered pair (j,k) except (2,3) and (3,4) (which could be tested together by computing t = P - r^3 for each prime r < P^(1/3) and, if t is square, checking whether sqrt(t) is a prime or the square of a prime) and those with k >= 5 (which could be tested by checking whether t = P - q^k is a prime power for each prime power q^k that is less than P and has k >= 5). (End)
a(17) <= 63985284333636413237490 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 37 * 41 * 43 * 59 * 61 * 103 * 409 = 10461281^3 + 250679912393^2. - Jon E. Schoenfield, Mar 31 2017

			a(1) = 13 = 2^2 + 3^2.
a(2) = 33 = 5^2 + 2^3 = 3 * 11.
a(3) = 130 = 3^2 + 11^2 = 2 * 5 * 13.
a(4) = 966 = 5^3 + 29^2 = 2 * 3 * 7 * 23.
a(5) = 14322 = 17^3 + 97^2 = 2 * 3 * 7 * 11 * 31.
a(6) = 81510 = 29^3 + 239^2 = 2 * 3 * 5 * 11 * 13 * 19.
a(7) = 3530730 = 41^4 + 89^3 = 2 * 3 * 5 * 7 * 17 * 23 * 43.
a(8) = 117535110 = 461^3 + 4423^2 = 2 * 3 * 5 * 7 * 11 * 17 * 41 * 73.
From _Jon E. Schoenfield_, Mar 14 2017: (Start)
a(9) = 2211297270 = 1301^3 + 3037^2 = 2 * 3 * 5 * 7 * 13 * 17 * 29 * 31 * 53.
a(10) = 131031070170 = 1361^3 + 358483^2 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 43 * 47 * 127. (End)
From _Giovanni Resta_, Mar 14 2017: (Start)
a(11) = 810571^2 + 8609^3,
a(12) = 12694849^2 + 13109^3. (End)
From _Jon E. Schoenfield_, Mar 18 2017: (Start)
a(13) = 24537703^2 + 140741^3.
a(14) = 639414679^2 + 178349^3.
a(15) = 1632727069^2 + 658649^3. (End)
a(16) = 1472015189^2 + 9094049^3. - _Jon E. Schoenfield_, Mar 19 2017

Cf. A225102, A246547.

N:= 1.2*10^8: # to get all terms <= N
PP:= {seq(seq(p^k,k=2..floor(log[p](N))), p = select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]))}:
PP:= sort(convert(PP,list)):
A:= 'A':
for i from 1 to nops(PP) do
  for j from 1 to i do
     Q:= PP[i]+PP[j];
     if Q > N then break fi;
     F:= ifactors(Q)[2];
     if max(seq(f[2],f=F))>1 then next fi;
     m:= nops(F);
     if not assigned(A[m]) or A[m] > Q then A[m]:= Q fi
od od:
seq(A[i],i=1..max(map(op,[indices(A)]))); # Robert Israel, Mar 01 2017

(* first 8 terms *) mx = 1.2*^8; a = 0 Range[8] + mx; p = Sort@ Flatten@ Table[ p^Range[2, Log[p, mx]], {p, Prime@ Range@ PrimePi@ Sqrt@ mx}]; Do[ j=1; While[j <= i && (v = p[[i]] + p[[j]]) < mx, f = FactorInteger@v; If[Max[Last /@ f] == 1, c = Length@f; If[c < 9 && v < a[[c]], a[[c]] = v]]; j++], {i, Length@p}]; a (* Giovanni Resta, Mar 19 2017 *)

do(lim)=my(v=List(),u=v,t,f); t=1; for(i=1,lim, t*=prime(i); if(t>lim,break); listput(v, oo)); v=Vec(v); for(e=2,logint(lim\=1,2), forprime(p=2,sqrtnint(lim-4,e), listput(u,p^e))); u=Set(u); for(i=1,#u, for(j=1,i, t=u[i]+u[j]; if(t>lim, break); f=factor(t)[,2]; if(vecmax(f)==1 && tif(k==oo,"?",k), v) \\ Charles R Greathouse IV, Mar 19 2017

do(lim)=my(v=List(),u=v,t,f,p2); t=1; for(i=1,lim, t*=prime(i); if(t>lim,break); listput(v, oo)); v=Vec(v); for(e=3,logint(lim\=1,2), forprime(p=2,sqrtnint(lim-4,e), listput(u,p^e))); u=Set(u); for(i=1,#u, for(j=1,i, t=u[i]+u[j]; if(t>lim, break); f=factor(t)[,2]; if(vecmax(f)==1 && tlim, break); f=factor(t)[,2]; if(vecmax(f)==1 && tlim, break); f=factor(t)[,2]; if(vecmax(f)==1 && tif(k==oo,"?",k), v) \\ Charles R Greathouse IV, Mar 19 2017

a(7)-a(8) from Giovanni Resta, Feb 21 2017
a(9)-a(10) from Jon E. Schoenfield, Mar 14 2017
a(11)-a(12) from Giovanni Resta, Mar 14 2017
a(13)-a(15) from Jon E. Schoenfield, Mar 18 2017
a(16) from Jon E. Schoenfield, Mar 19 2017

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A225103 Numbers that can be represented as a sum of two distinct nontrivial prime powers in two or more ways.

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A282550 Perfect powers that are the sum of two distinct proper prime powers (A246547).

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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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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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A282578 Least k such that k^n is the sum of two distinct proper prime powers (A246547), or 0 if no such k exists.

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A282686 Least sum of two proper prime powers (A246547) that is the product of n distinct primes.

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