A076467 -id:A076467 - cp's OEIS frontend

A304434 Numbers n such that n^4 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).

3, 5, 6, 9, 10, 12, 13, 14, 15, 17, 20, 24, 25, 26, 28, 29, 30, 34, 35, 36, 37, 39, 40, 41, 42, 45, 48, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 65, 68, 70, 71, 72, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 96, 97, 98, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113

Offset: 1

M. F. Hasler, May 22 2018

nonn

Motivated by the search of solutions to a^n + b^(2n+2)/4 = (perfect square), which arises when searching solutions to x^n + y^(n+1) = z^(n+2) of the form x = a*z, y = b*z. It turns out that many solutions are of the form a^n = d (b^(n+1) + d), where d is a perfect power.

			3^4 = 2^5 + 7^2; 5^4 = 7^2 + 24^2, ...

Robert Israel, Table of n, a(n) for n = 1..1301

Cf. A304433, A001597 (perfect powers), A076467 (third or higher powers).

N:= 200: # to get terms <= N
N4:= N^4:
P:= {seq(seq(x^k,k=3..floor(log[x](N4))),x=2..floor(N4^(1/3)))}:
filter:= proc(n) local n4, Pp;
  n4:= n^4;
  if remove(t -> subs(t,x)<=1 or subs(t,y)<=1 or subs(t,x-y)=0, [isolve(x^2+y^2=n4)]) <> [] then return true fi;
  Pp:= map(t ->n4-t, P minus {n4, n4/2});
  (Pp intersect P <> {}) or (select(issqr,Pp) <> {})
end proc:
A:= select(filter, [$2..N]); # Robert Israel, May 24 2018

L=200^4; P=List(); for(x=2, sqrtnint(L,3), for(k=3, logint(L, x), listput(P, x^k))); #P=Set(P) \\ This P = A076467 \ {1} = A111231 \ {0} up to limit L.
is_A304434(n)={for(i=1, #s=sum2sqr(n=n^4), vecmin(s[i])>1 && s[i][1]!=s[i][2] && return(1)); for(i=1, #P, n>P[i]||return; ispower(n-P[i])&& P[i]*2 != n && return(1))} \\ The above P must be computed up to L >= n^4. For sum2sqr() see A133388.

A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

Original entry on oeis.org

4, 9, 25, 225, 676, 2116, 6724, 7921, 8100, 16641, 104329, 131044, 160801, 176400, 372100, 389376, 705600, 4096576, 7306209, 7884864, 47444544, 146385801, 254817369, 373262400, 607622500, 895804900, 1121580100, 1330936324, 1536875209, 2097182025, 2258435529, 2749953600

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

Apparently, all known terms (checked through 10^18) are squares with maximum exponent 2, i.e., terms of A111245 (squares that are not a higher power). This would imply that of 3 immediately adjacent perfect powers, at least one is a term of A111245. Is there a known counterexample of 3 consecutive perfect powers, none of which is in A111245?

			The first terms, assuming 1 being at least a cube:
.
  n   p1  x^p1  p2  a(n)  p3  z^p3
                   =y^p2
  1  >2     1   2     4   3     8
  2   3     8   2     9   4    16
  3   4    16   2    25   3    27
  4   3   216   2   225   5   243
  5   4   625   2   676   6   729

Hugo Pfoertner, Table of n, a(n) for n = 1..181 (terms < 10^18)

Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.

a340642(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2>2&p0>2, print1(n1,", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340642(50000000)

A076468 Perfect powers m^k where k >= 4.

Original entry on oeis.org

1, 16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 1296, 2048, 2187, 2401, 3125, 4096, 6561, 7776, 8192, 10000, 14641, 15625, 16384, 16807, 19683, 20736, 28561, 32768, 38416, 46656, 50625, 59049, 65536, 78125, 83521, 100000, 104976, 117649

Offset: 1

Robert G. Wilson v, Oct 14 2002

nonn

If p|n then at least p^4|n.

Subsequence of A036967. - R. J. Mathar, May 27 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001597, A036967, A076467, A076469, A076470.

Cf. A002117, A013661.

import qualified Data.Set as Set (null)
import Data.Set (empty, insert, deleteFindMin)
a076468 n = a076468_list !! (n-1)
a076468_list = 1 : f [2..] empty where
   f xs'@(x:xs) s | Set.null s || m > x ^ 4 = f xs $ insert (x ^ 4, x) s
                  | m == x ^ 4  = f xs s
                  | otherwise = m : f xs' (insert (m * b, b) s')
                  where ((m, b), s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 19 2013

a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 3, a = Append[a, n]; Print[n]], {n, 2, 131071}]; a

from sympy import mobius, integer_nthroot
def A076468(n):
    def f(x): return int(n+2+x-integer_nthroot(x,4)[0]-(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,9)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]+integer_nthroot(x,3*k)[0]-3) for k in range(5,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Sum_{n>=1} 1/a(n) = 3 - zeta(2) - zeta(3) + Sum_{k>=2} mu(k)*(3 - zeta(k) - zeta(2*k) - zeta(3*k)) = 1.1473274274... . - Amiram Eldar, Dec 03 2022

A076469 Perfect powers m^k where k >= 5.

Original entry on oeis.org

1, 32, 64, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125, 4096, 6561, 7776, 8192, 15625, 16384, 16807, 19683, 32768, 46656, 59049, 65536, 78125, 100000, 117649, 131072, 161051, 177147, 248832, 262144, 279936, 371293, 390625, 524288, 531441

Offset: 1

Robert G. Wilson v, Oct 14 2002

nonn

If p|n when at least p^5|n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001597, A076467, A076468, A076470.

Cf. A002117, A013661, A013662.

a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 4, a = Append[a, n]; Print[n]], {n, 2, 537823}]; a

from sympy import mobius, integer_nthroot
def A076469(n):
    def f(x): return int(n+3+x-(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,8)[0]-integer_nthroot(x,9)[0]-integer_nthroot(x,12)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]+integer_nthroot(x,3*k)[0]+integer_nthroot(x,k<<2)[0]-4) for k in range(5,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Sum_{n>=1} 1/a(n) = 4 - zeta(2) - zeta(3) - zeta(4) + Sum_{k>=2} mu(k)*(4 - zeta(k) - zeta(2*k) - zeta(3*k) - zeta(4*k)) = 1.06932853458... . - Amiram Eldar, Dec 03 2022

A226777 Higher powers that are sums of two distinct higher powers.

Original entry on oeis.org

243, 2744, 6561, 177147, 185193, 474552, 614656, 810000, 941192, 1124864, 1419857, 1500625, 3241792, 4782969, 7962624, 11239424, 16003008, 17850625, 21952000, 26873856, 28372625, 52200625, 68574961, 82312875, 117649000, 129140163, 162771336, 200201625, 238328000

Offset: 1

Robert Israel, Jun 17 2013

nonn

x is in the sequence iff there are distinct y,z such that x = y + z and x,y,z are all in A076467.

			243 is in the sequence because 243 = 3^5 = 3^3 + 6^3.

Robert Israel and Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 (first 264 terms from Robert Israel)

import qualified Data.Set as Set (null, split, filter)
import Data.Set (Set, empty, insert, member)
a226777 n = a226777_list !! (n-1)
a226777_list = f a076467_list empty where
   f (x:xs) s | Set.null $ Set.filter ((`member` s) . (x -)) s'
                          = f xs (x `insert` s)
              | otherwise = x : f xs (x `insert` s)
              where (s', _) = Set.split (x `div` 2) s
-- Reinhard Zumkeller, Sep 13, Jun 19 2013

N :=  10^12: # to get terms up to N
S := {seq(seq(a^x, a=1 .. floor(N^(1/x))), x = 3 .. floor(log[2](N)))}:
f:= proc(n) local L; L:= S[1..n-1] minus {S[n]/2}; nops(map2(`-`,S[n],L) intersect L) > 0 end proc;
A:= map(t -> S[t], select(f,[$1..nops(S)]));

max = 3*10^8; pp = Join[{1}, Table[n^k, {k, 3, Floor[Log[2, max]]}, {n, 2, Floor[max^(1/k)]}] // Flatten // Union]; Select[Total /@ Subsets[pp, {2}], MemberQ[pp, #]&] // Union (* Jean-François Alcover, Feb 14 2018 *)

A261782 Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2.

Original entry on oeis.org

16, 32, 64, 128, 243, 256, 512, 1024, 2048, 2744, 4096, 6561, 8192, 16384, 32768, 65536, 131072, 177147, 185193, 262144, 474552, 524288, 614656, 810000, 941192, 1048576, 1124864, 1419857, 1500625, 2097152, 3241792, 4194304

Offset: 1

Anatoly E. Voevudko, Aug 31 2015

nonn

Beal's conjecture states that A, B, and C have a common prime factor.

Theorem. If A, B are odd and x, y are even, Beal's conjecture has no counterexample. Proof: Let D be odd, D > 1 and let w be even, w > 2. Then D^w == 9 (mod 24) while D == 0 (mod 3); otherwise, D^w == 1 (mod 24) (trivial). Any even C^z == {0; 8; 16} (mod 24): if C == 0 (mod 3), C^z == 0 (mod 24); if C == 1 (mod 3), C^z == 16 (mod 24); if C == 2 (mod 3), C^z == 8 (mod 24), while z is odd, and C^z == 16 (mod 24), while z is even (trivial). But C^z == (x'+y') (mod 24) where A^x = x' (mod 24), B^y = y' (mod 24); since (x'+y') = {2; 10; 18}, C^z == {2; 10; 18} (mod 24), which cannot be a counterexample to Beal's conjecture. - Sergey Pavlov, May 08 2021

			2^3 + 2^3 = 2^4 = 16, so 16 is in the sequence.

Anatoly E. Voevudko and Charles R Greathouse IV, Table of n, a(n) for n = 1..1229 (first 196 terms from Voevudko)
American Mathematical Society, Beal Prize
Anatoly E. Voevudko, Description of all powers in b245713
Anatoly E. Voevudko, Description of all powers in b261782
Wikipedia, Beal's conjecture

Subsequence of A076467.

Cf. A245713.

is(n)=if(ispower(n)<3, return(0)); for(x=3,logint((n+1)\2,2), for(A=2,sqrtnint(n,x), if(ispower(n-A^x)>2, return(1)))); 0 \\ Charles R Greathouse IV, Sep 03 2015

list(lim)=my(v=List(),u=v,t); for(z=3,logint(lim\=1,2), for(C=2,sqrtnint(lim,z), listput(v,C^z))); v=Set(v); for(i=1,#v, for(j=i,#v, t=v[i]+v[j]; if(t>lim, break); if(setsearch(v,t), listput(u,t)))); Set(u) \\ Charles R Greathouse IV, Sep 03 2015

A330980 a(n) = (p1 + p2)/216 such that p1 >= 5 and p2 = p1 + 2 are twin primes and p1 + p2 is a k-th power with k >= 3.

Original entry on oeis.org

1, 1296, 24389, 274625, 531441, 970299, 2343750, 2515456, 4492125, 5268024, 5451776, 6967871, 8000000, 18821096, 25672375, 27270901, 32461759, 37748736, 41421736, 43243551, 50653000, 64000000, 69426531, 80062991, 81746504, 82881856, 94818816, 100663296

Offset: 1

Hugo Pfoertner, Jan 05 2020

nonn

The values of k corresponding to the first terms are: 3, 7, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, ...

			a(1) = 1: p1 = 107 and p2 = 109 is the first pair with a sum that is a 3rd power, 216=6^3;
a(2) = 1296: p1 = 1296*108 - 1 = 139967, p2 = 1296*108 + 1 = 139969, p1 + p2 = 279936 = 6^7.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A076467, A119768, A270231, A330978.

my(pp=5,j); forprime(p=7,10000000000, if(p-pp==2, if(j=ispower(p+pp), if(j>2, print1((p+pp)/216,", ")))); pp=p)

A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

Original entry on oeis.org

1, 8, 16, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 343, 350, 360, 375, 378, 384, 392, 400, 405, 420, 432, 441, 448, 450, 480, 486, 490, 500, 504, 512, 525

Offset: 1

Labos Elemer, Nov 21 2003

What is the asymptotic growth of this sequence?

Answer: a(n) ~ k*n, where k = 1/A175475. That is, about 4.8% of numbers are in this sequence. - Charles R Greathouse IV, Jul 14 2014

			378 = 2 * 3^3 * 7 is a term of the sequence since 7 < 7.23... = 378^(1/3).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A063763, A001248, A048098, A036966, A054744, A059172, A068936, A076405, A076467, A006530, A175475.

filter:= n ->
evalb(max(seq(f[1],f=ifactors(n)[2]))^3 <= n):
select(filter, [$1..1000]); # Robert Israel, Jul 14 2014

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; Do[If[ !Greater[ma[n], gy=n^(1/3)//N]&&!PrimeQ[n], Print[n(*, {gy, ma[n]}*)]], {n, 1, 1000}]
Select[Range[1000], (FactorInteger[#][[-1,1]])^3 <= # &] (* T. D. Noe, Sep 14 2011 *)
Select[Range[1000],FactorInteger[#][[-1,1]]<=CubeRoot[#]&] (* Harvey P. Dale, Jun 30 2025 *)

is(n)=my(f=factor(n)[,1]);f[#f]^3<=n \\ Charles R Greathouse IV, Sep 14 2011

from sympy import primefactors
def ok(n):
    if n==1 or max(primefactors(n))**3<=n: return True
    else: return False
print([n for n in range(1, 1001) if ok(n)]) # Indranil Ghosh, Apr 23 2017

Solutions to A006530(n) <= n^(1/3).

A340585 Noncube perfect powers.

Original entry on oeis.org

4, 9, 16, 25, 32, 36, 49, 81, 100, 121, 128, 144, 169, 196, 225, 243, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2048, 2116, 2187, 2209, 2304, 2401, 2500

Offset: 1

Hugo Pfoertner, Jan 12 2021

nonn

This was the original definition of A239870. However, the true definition of that sequence seems to be slightly different.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000578, A001597, A002117, A007412, A072102, A076467, A097054, A239728.

filter:= proc(n) local g;
  g:= igcd(op(ifactors(n)[2][..,2]));
  g > 1 and (g mod 3 <> 0)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jan 12 2021

Select[Range[2, 2500], (g = GCD @@ FactorInteger[#][[;; , 2]]) > 1 && !Divisible[g, 3] &] (* Amiram Eldar, Jan 12 2021 *)

for(n=2,2500,if( ispower(n) % 3, print1(n,", ")))

from math import isqrt
from sympy import mobius, integer_nthroot
def A340585(n):
    def f(x): return int(n+x-isqrt(x)+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(5,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Sum_{n>=1} 1/a(n) = 1 - zeta(3) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 1 - A002117 + A072102 = 0.6724074652... - Amiram Eldar, Jan 12 2021

A226950 Higher powers having partitions into distinct higher powers in more than one way.

Original entry on oeis.org

7449150177, 8936757492481, 11587386200625, 22449633661000, 30511719124992, 36443545848801, 283680450809856, 583096733580816, 613579106939841, 3958783819215057, 4098048384032001, 4556608567054336, 13469350037585841, 23887131799781376, 36604958689202176, 58634065908167841, 69952404620958561, 91953699475456000, 124976001535967232, 149272763796688896, 183001280170947121, 225430653627891712

Offset: 1

Robert Israel and Reinhard Zumkeller, Sep 13 2013

nonn

A power m^k is called a higher power if k > 2, cf. A076467.

			a(1) = 7449150177 = 1953^3 = 1116^3 + 279^4 = 217^4 + 1736^3;
a(2) = 8936757492481 = 1729^4 = 1729^3 + 20748^3 = 15561^3 + 17290^3;
see link for more examples and more info.

Reinhard Zumkeller, First 22 terms of A226950

Cf. A051388, subsequence of A226777.

import qualified Data.Set as Set (split, filter)
import Data.Set (Set, empty, size, insert, member)
a226950 n = a226950_list !! (n-1)
a226950_list = f a076467_list empty where
   f (x:xs) s | size s'' <= 1 = f xs (x `insert` s)
              | otherwise     = x : f xs (x `insert` s)
              where s'' = Set.filter ((`member` s) . (x -)) s'
                    (s', _) = Set.split (x `div` 2) s

cp's OEIS Frontend

A304434 Numbers n such that n^4 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).

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A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

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A076468 Perfect powers m^k where k >= 4.

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A076469 Perfect powers m^k where k >= 5.

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A226777 Higher powers that are sums of two distinct higher powers.

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A261782 Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2.

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A330980 a(n) = (p1 + p2)/216 such that p1 >= 5 and p2 = p1 + 2 are twin primes and p1 + p2 is a k-th power with k >= 3.

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A090081 Cube root-smooth numbers: numbers k whose largest prime factor does not exceed the cube root of k.

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