A076467 -id:A076467 - cp's OEIS frontend

A340701 Upper of a pair of adjacent perfect powers, both with exponents > 2.

32, 81, 128, 256, 1024, 1331, 2197, 50653, 59319, 195112, 279936, 456976, 614656, 3112136, 6436343, 22667121, 25412184, 38958219, 62748517, 96071912, 131096512, 418195493, 506261573, 741217625, 796597983, 1249243533, 2136750625, 2217373921, 5554637011, 5802888573

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

The corresponding lower members of the pairs are A340700.

Cf. A001597, A076467, A340642, A340702, A340703, A340704, A340705, A340706.

a(n) = A340703(n)^A340705(n) = A340700(n) + A340706(n).

A340702 Root of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

3, 2, 5, 3, 10, 6, 3, 15, 3, 21, 23, 77, 85, 42, 186, 283, 71, 79, 89, 99, 107, 143, 150, 165, 168, 188, 1288, 1304, 273, 276, 1858, 2542, 2685, 396, 435, 4246, 612, 5619, 6109, 710, 2, 6549, 6573, 199, 201, 7082, 7563, 7888, 855, 7, 938, 11562, 1211, 1312, 1438

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

Cf. A001597, A076467, A340700, A340701, A340703, A340704, A340705, A340706.

A340700(n) = a(n)^A340704(n).

A340703 Root of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

2, 3, 2, 2, 2, 11, 13, 37, 39, 58, 6, 26, 28, 146, 23, 69, 294, 339, 13, 458, 508, 53, 797, 905, 927, 1077, 215, 217, 1771, 1797, 283, 358, 373, 2908, 3296, 526, 5196, 649, 691, 6334, 6502, 728, 730, 6783, 6897, 772, 811, 837, 8115, 8786, 9182, 1115, 12908, 14363

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

Cf. A001597, A076467, A340700, A340701, A340702, A340704, A340705, A340706.

A340701(n) = a(n)^A340705(n).

A340704 Exponent of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

3, 6, 3, 5, 3, 4, 7, 4, 10, 4, 4, 3, 3, 4, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 3, 3, 3, 4, 4, 3, 4, 3, 3, 4, 38, 3, 3, 5, 5, 3, 3, 3, 4, 14, 4, 3, 4, 4, 4, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 3

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

Cf. A001597, A076467, A340700, A340701, A340702, A340703, A340705, A340706.

A340700(n) = A340702(n)^a(n).

A340705 Exponent of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

5, 4, 7, 8, 10, 3, 3, 3, 3, 3, 7, 4, 4, 3, 5, 4, 3, 3, 7, 3, 3, 5, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 5, 3, 4, 4, 3, 4, 4, 4, 4, 4, 5, 4, 5, 4, 3, 4, 4, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 4

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

			See A340700.

Hugo Pfoertner, Table of n, a(n) for n = 1..1670

Cf. A001597, A076467, A340700, A340701, A340702, A340703, A340704, A340706.

A340701(n) = A340703(n)^a(n).

A340706 Difference between upper and lower member of a pair of adjacent perfect powers A340700 and A340701, both with exponents > 2.

Original entry on oeis.org

5, 17, 3, 13, 24, 35, 10, 28, 270, 631, 95, 443, 531, 440, 1487, 1934, 503, 8138, 6276, 12311, 16911, 33892, 11573, 17000, 3807, 45197, 30753, 31457, 65170, 105597, 127209, 206808, 109516, 139456, 377711, 530040, 561600, 690742, 952332, 457704, 671064, 353107

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

The differences are expected to be bounded below by the Lang-Waldschmidt conjecture (see Waldschmidt 2013, p. 6, Conjecture 6).

			See A340700.

Hugo Pfoertner, Table of n, a(n) for n = 1..1670
Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.

Cf. A001597, A076467, A340700, A340701, A340702, A340703, A340704, A340705.

a(n) = A340701(n) - A340700(n).

A076470 Perfect powers m^k where k >= 6.

Original entry on oeis.org

1, 64, 128, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 46656, 59049, 65536, 78125, 117649, 131072, 177147, 262144, 279936, 390625, 524288, 531441, 823543, 1000000, 1048576, 1594323, 1679616, 1771561

Offset: 1

Robert G. Wilson v, Oct 14 2002

nonn

A necessary but not sufficient condition is that if p|n when at least p^6|n.

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

Cf. A001597, A076467, A076468, A076469.
Different from A069493.
Cf. A002117, A013661, A013662, A013663.

a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 4, a = Append[a, n]; Print[n]], {n, 2, 1953124}]; a

from sympy import mobius, integer_nthroot
def A076470(n):
    def f(x): return int(n+9+x-(sum(integer_nthroot(x,d)[0] for d in (6,10,15))<<1)-sum(integer_nthroot(x,d)[0] for d in (8,9,12,20,25))+sum(mobius(k)*(sum(integer_nthroot(x,k*i)[0] for i in range(1,6))-5) for k in range(6,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) = 5 - zeta(2) - zeta(3) - zeta(4) - zeta(5) + Sum_{k>=2} mu(k)*(5 - zeta(k) - zeta(2*k) - zeta(3*k) - zeta(4*k) - zeta(5*k)) = 1.03342597171... . - Amiram Eldar, Dec 03 2022

A111231 Numbers which are perfect powers m^k equal to the sum of m distinct primes.

Original entry on oeis.org

0, 8, 16, 27, 32, 64, 81, 125, 128, 216, 243, 256, 343, 512, 625, 729, 1000, 1024, 1296, 1331, 1728, 2048, 2187, 2197, 2401, 2744, 3125, 3375, 4096, 4913, 5832, 6561, 6859, 7776, 8000, 8192, 9261, 10000, 10648, 12167, 13824, 14641, 15625, 16384, 16807

Offset: 1

Giovanni Teofilatto, Oct 28 2005

nonn

Perfect powers m^k with k >= 3, m = 0 or m > 1.
Is a(n) = A076467(n) for all n > 1? - R. J. Mathar, May 22 2009
A sum of m distinct primes is >= A007504(m) ~ m^2(log m)/2 > m^2, also for small m, therefore the second condition excludes squares m^2. On the other hand, considering results related to Goldbach's conjecture (e.g., every even number >= 4 is the sum of at most 4 primes), it is increasingly improbable that some m^k with k >= 3 is not the sum of m primes. This explains the first comment - but can it be rigorously proved? - M. F. Hasler, May 25 2018

			a(1) = 0 because 0 = 0^2 = 0^3 is the sum of 0 primes;
a(2) = 8 because 8 = 2^3 = 3 + 5, sum of 2 primes;
a(3) = 16 because 16 = 2^4 = 3 + 13, sum of 2 primes.
a(4) = 27 because 27 = 3^3 = 3 + 11 + 13, sum of 3 primes.

Cf. A007504, A076467.

is(n,d)={if(d=ispower(n), fordiv(d,e,e>1&&forvec(v=vector(d=sqrtnint(n,e)-1,i,[1,primepi((n-1)\2-d+3)]),prime(v[#v])<(d=n-vecsum(apply(i->prime(i),v)))&&isprime(d)&&return(1),2)), !n)} \\ M. F. Hasler, May 25 2018

Offset corrected by R. J. Mathar, May 25 2009

Edited by M. F. Hasler, May 25 2018

A245713 Sorted imperfect powers b^p with b > 0, p > 2, with multiplicity.

Original entry on oeis.org

1, 8, 16, 27, 32, 64, 64, 81, 125, 128, 216, 243, 256, 256, 343, 512, 512, 625, 729, 729, 1000, 1024, 1024, 1296, 1331, 1728, 2048, 2187, 2197, 2401, 2744, 3125, 3375, 4096, 4096, 4096, 4096, 4913, 5832, 6561, 6561, 6859, 7776, 8000, 8192, 9261

Offset: 1

Anatoly E. Voevudko, Jul 30 2014

No multiple terms for b=1.
This sequence strictly follows requirements of the Beal conjecture.
Less than 550 of these powers satisfy 196 Beal's conjecture equations.

Anatoly E. Voevudko, Table of n, a(n) for n = 1..11539
American Mathematical Society, Beal Prize
Alf van der Poorten, Remarks on the sequence of 'perfect' numbers
Anatoly E. Voevudko, Description of all powers in b245713
Eric W. Weisstein, World of Mathematics, Perfect Power
Wikipedia, Beal's conjecture

Cf. A001597, A023057, A072103, A076467.

N:= 10^5: # to get all terms <= N
L:= [1, seq(seq(b^p, p=3..floor(log[b](N))),b=2..floor(N^(1/3)))]:
sort(L); # Robert Israel, Nov 09 2015

mx = 10000; Join[{1}, Sort@ Flatten@ Table[b^p, {b, 2, Sqrt@ mx}, {p, 3, Log[b, mx]}]] (* Robert G. Wilson v, Nov 09 2015 *)

A245713(lim)={my(L=List(1),lim2=logint(lim,2));for(p=3,lim2, for(b=2,sqrtnint(lim,p),listput(L, b^p);));listsort(L); print(L);} \\ Anatoly E. Voevudko, Sep 21 2015

A304433 Numbers n such that n^3 is the sum of two distinct perfect powers > 1 (x^k + y^m; x, y, k, m >= 2).

Original entry on oeis.org

5, 7, 8, 10, 12, 13, 14, 17, 20, 25, 26, 28, 29, 32, 33, 34, 37, 40, 41, 45, 48, 50, 52, 53, 56, 57, 58, 61, 63, 65, 68, 71, 72, 73, 74, 78, 80, 82, 85, 89, 90, 97, 98, 100, 101, 104, 105, 106, 109, 112, 113, 114, 116, 117, 122, 125, 126, 128

Offset: 1

M. F. Hasler, May 12 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.

			5^3 = 125 = 4^2 + 11^2; 7^3 = 10^2 + 3^5; 8^3 = 13^2 + 7^3, ...

Robert Israel, Table of n, a(n) for n = 1..5970

Cf. A001597 (perfect powers), A076467 (cubes and higher powers), A304434, A304435, A304436 (analog for n^4, n^5, n^6).

N:= 200: # to get terms <= N
N3:= N^3:
P:= {seq(seq(x^k, k=3..floor(log[x](N3))), x=2..N)}:
filter:= proc(n) local n3, Pp, x, y;
  n3:= n^3;
  if remove(t -> subs(t, x)<=1 or subs(t, y)<=1 or subs(t, x-y)=0, [isolve(x^2+y^2=n3)]) <> [] then return true fi;
  Pp:= map(t ->n3-t, P minus {n3, n3/2});
   (Pp intersect P <> {}) or (select(issqr, Pp) <> {})
end proc:
select(filter, [$2..N]); # Robert Israel, Jun 01 2018

M = 200;
M3 = M^3;
P = Union@ Flatten@ Table[Table[x^k, {k, 3, Floor[Log[x, M3]]}], {x, 2, M}];
filterQ[n_] := Module[{n3, Pp, x, y}, n3 = n^3; If[Solve[x > 1 && y > 1 && x != y && x^2 + y^2 == n3, {x, y}, Integers] != {}, Return[True]]; Pp = n3 - (P ~Complement~ {n3, n3/2}); (Pp ~Intersection~ P) != {} || Select[ Pp, IntegerQ[Sqrt[#]]&] != {}];
Select[Range[2, M], filterQ] (* Jean-François Alcover, Jun 21 2020, after Robert Israel *)

L=200^3; 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(n,e=3)={for(i=1, #s=sum2sqr(n=n^e), 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^3. For sum2sqr() see A133388.

cp's OEIS Frontend

A340701 Upper of a pair of adjacent perfect powers, both with exponents > 2.

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A340702 Root of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

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A340703 Root of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

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A340704 Exponent of the lower member A340700 of a pair of adjacent perfect powers, both with exponents > 2.

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A340705 Exponent of the upper member A340701 of a pair of adjacent perfect powers, both with exponents > 2.

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A340706 Difference between upper and lower member of a pair of adjacent perfect powers A340700 and A340701, both with exponents > 2.

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A076470 Perfect powers m^k where k >= 6.

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A111231 Numbers which are perfect powers m^k equal to the sum of m distinct primes.

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A245713 Sorted imperfect powers b^p with b > 0, p > 2, with multiplicity.

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Maple