A036966 -id:A036966 - cp's OEIS frontend

A363177 Primitive abundant numbers (A071395) that are cubefull numbers (A036966).

26376098024367, 33912126031329, 1910383099764867, 2792098376579421, 5229860083034911875, 6886512413632368153, 8815747507513708671, 28966027524687899919, 42200802302982406288, 89594138836162749375, 224439112362213402759, 288564573037131517833, 512767531125033485625

Offset: 1

Amiram Eldar, May 19 2023

nonn

It seems that this sequence is also the intersection of A036966 and A091191 (checked up to 10^27).

Are there terms that are 4-full numbers (A036967)? There are none below 10^27.

Amiram Eldar, Table of n, a(n) for n = 1..42 (terms below 10^27)
Wikipedia, Primitive abundant number.

Intersection of A036966 and A071395.
Subsequence of A363169 and A363175.
A306797 is a subsequence.
Cf. A036967, A091191.

A210470 Powerful numbers (A001694) which can be written as the sum of two relatively prime 3-powerful numbers (A036966) different from 1.

Original entry on oeis.org

841, 968, 2312, 3528, 5041, 5776, 12769, 14884, 16641, 45125, 51984, 109561, 123823, 157609, 168921, 207576, 373321, 450241, 498436, 609725, 711828, 731025, 798768, 940896, 1223048, 1590121, 1792921, 2478843, 2481992, 2526752, 3157729, 3964081, 5346675, 6255001

Offset: 1

N. J. A. Sloane, Apr 22 2013

nonn

			841 = 216+625 ; 968 = 343+625 ; 2312=125+2187;

Jean-Marie de Konninck, Those Fascinating Numbers, Amer. Math. Soc., 2009.
Alonso Del Arte, Posting to the Sequence Fans Mailing List, Mar 10 2011.

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

Cf. A001694, A036966.

isA210470 := proc(n)
    if isA001694(n) then
        for i from 2 do
            p3 := A036966(i) ;
            if p3+2 > n then
                return false;
            end if;
            p3comp := n-p3 ;
            if isA036966(p3comp) and igcd(p3,p3comp) = 1 then
                # print(n,p3,p3comp) ;
                return true;
            end if;
        end do:
        return false;
    else
        return false;
    end if;
end proc:
for n from 1 do
    if isA210470(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 01 2013

With[{max = 10^7}, powQ[n_, e_] := Min[FactorInteger[n][[;; , 2]]] > e; pows = Union[Flatten[Table[i^2*j^3, {j, max^(1/3)}, {i, Sqrt[max/j^3]}]]]; Select[Union[Plus @@@ Select[Tuples[Select[pows, powQ[#, 2] &], {2}], CoprimeQ @@ # &]], # < max && powQ[#, 1] &]] (* Amiram Eldar, Jan 30 2023 *)

{ a in A001694: a=b+c and b,c >1 and b,c in A036966 and gcd(b,c)=1}. - R. J. Mathar, May 01 2013

More terms from Amiram Eldar, Jan 30 2023

A362972 Squarefree kernels of cubefull numbers (A036966).

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 5, 2, 6, 3, 2, 7, 6, 2, 5, 6, 3, 6, 10, 2, 6, 11, 6, 6, 10, 2, 3, 13, 7, 6, 14, 5, 15, 6, 6, 10, 2, 17, 10, 6, 14, 6, 3, 19, 6, 6, 10, 2, 21, 10, 15, 6, 22, 14, 6, 23, 6, 11, 6, 5, 10, 2, 7, 15, 6, 26, 14, 3, 10, 6, 22, 14, 6, 29, 10, 30, 6

Offset: 1

Amiram Eldar, May 13 2023

nonn

			A036966(2) = 8 = 2^3, therefore a(2) = 2.
A036966(10) = 216 = 2^3 * 3^2, therefore a(10) = 2 * 3 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, The kernel of powerful numbers, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, eq. (18), p. 728.

Cf. A007947, A036966, A084371, A362974.

seq[kmax_] := Module[{s = {1}}, Do[f = FactorInteger[k]; If[Min@f[[;; , 2]] > 2, AppendTo[s, Times @@ f[[;; , 1]]]], {k, 2, kmax}]; s]; seq[10^5]

lista(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(k==1 || vecmin(f[, 2]) > 2, print1(vecprod(f[, 1]), ", ")));}

a(n) = A007947(A036966(n)).
Sum_{A036966(k) < x} a(k) = c * x^(2/3) + o(x^(2/3)), where c = (3/Pi^2) * Product_{p prime} (1 + 1/((p+1)*(p^(2/3)-1))) = 0.7356919531... (Jakimczuk, 2017). [corrected Sep 21 2024]
Sum_{k=1..n} a(k) ~ (c / A362974 ^ 2) * n^2, where c is the constant above.

A363013 a(n) is the number of prime factors (counted with multiplicity) of the n-th cubefull number (A036966).

Original entry on oeis.org

0, 3, 4, 3, 5, 6, 4, 3, 7, 6, 5, 8, 3, 7, 9, 4, 7, 6, 8, 6, 10, 8, 3, 9, 8, 7, 11, 7, 3, 4, 9, 6, 5, 6, 10, 9, 8, 12, 3, 7, 10, 7, 9, 8, 3, 11, 10, 9, 13, 6, 8, 7, 11, 6, 8, 10, 3, 12, 4, 11, 6, 10, 14, 5, 7, 10, 6, 7, 9, 9, 12, 7, 9, 11, 3, 8, 9, 13, 7, 4, 3

Offset: 1

Amiram Eldar, May 13 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

Cf. A001222, A036966, A362974.

Similar sequences: A072047, A076399, A360729.

PrimeOmega[Select[Range[10000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

iscubefull(n) = n==1 || vecmin(factor(n)[, 2]) > 2;
apply(bigomega, select(iscubefull, [1..10000]))

a(n) = A001222(A036966(n)).
a(n) >= 3, for n > 1.
Sum_{A036966(k) < x} a(k) = 3*c*x^(1/3)*log(log(x)) + (3*(B_2 - log(2)) + Sum_{p prime} ((4*p^(1/3)+5)/(p^(5/3)+p^(1/3)+1)))*c*x^(1/3) + O(x^(1/3)/sqrt(log(x))), where B_2 = A083342 and c = A362974 (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

A363014 Cubefull numbers (A036966) with a record gap to the next cubefull number.

Original entry on oeis.org

1, 8, 16, 32, 81, 128, 343, 512, 729, 864, 1024, 1331, 3456, 4096, 6912, 8192, 12167, 25000, 32768, 35937, 43904, 46656, 55296, 70304, 93312, 110592, 117649, 140608, 186624, 287496, 331776, 357911, 373248, 592704, 707281, 889056, 1000000, 1124864, 1157625, 1296000

Offset: 1

Amiram Eldar, May 13 2023

nonn

This sequence is infinite since the asymptotic density of the cubefull numbers is 0.

The corresponding record gaps are 7, 8, 16, 49, 47, 215, 169, 217, 135, 160, ... .

			The sequence of cubefull numbers begins with 1, 8, 16, 27, 32, 64, 81 and 125. The differences between these terms are 7, 8, 11, 5, 32, 17 and 44. The record values, 7, 8, 11, 32 and 44 occur after the cubefull numbers 1, 8, 16, 32 and 81, the first 5 terms of this sequence.

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

Cf. A036966, A349062.

cubQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 2; seq[kmax_] := Module[{s = {}, k1 = 1, gapmax = 0, gap}, Do[If[cubQ[k], gap = k - k1; If[gap > gapmax, gapmax = gap; AppendTo[s, k1]]; k1 = k], {k, 2, kmax}]; s]; seq[10^6]

iscubefull(n) = n==1 || vecmin(factor(n)[, 2]) > 2;
lista(kmax) = {my(gapmax = 0, gap, k1 = 1); for(k = 2, kmax, if(iscubefull(k), gap = k - k1; if(gap > gapmax, gapmax = gap; print1(k1, ", ")); k1 = k));}

A307703 Highly powerful numbers (A005934) that are not cubeful (A036966).

Original entry on oeis.org

4, 144, 288, 86400, 129600, 194400, 259200, 518400, 190512000, 317520000, 381024000, 635040000, 9681819840000, 215982036990720000, 9466852651364908800000, 14200278977047363200000, 28400557954094726400000, 174294224164279335916800000, 522882672492838007750400000

Offset: 1

Amiram Eldar, Apr 22 2019

Lacampagne and Selfridge proved that these are the only terms.

The positions of the terms in A005934 are 2, 8, 10, 25, 27, 28, 30, 33, 55, 58, 60, 62, 107, 161, 230, 234, 240, 302, 315.

Carole B. Lacampagne and John L. Selfridge, Large highly powerful numbers are cubeful, Proceedings of the American Mathematical Society, Vol. 91, No. 2 (1984), pp. 173-181.

Cf. A005934, A036966.

pmax = 1; s = {}; Do[e = FactorInteger[n][[;; , 2]]; p = Times @@ e; If[p > pmax, pmax = p; If[Min[e] < 3, AppendTo[s, n]]], {n, 2, 10^6}]; s

A030078 Cubes of primes.

Original entry on oeis.org

8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389, 29791, 50653, 68921, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091, 2571353, 2685619, 3307949

Offset: 1

Patrick De Geest

Numbers with exactly three factorizations: A001055(a(n)) = 3 (e.g., a(4) = 1*343 = 7*49 = 7*7*7). - Reinhard Zumkeller, Dec 29 2001
Intersection of A014612 and A000578. Intersection of A014612 and A030513. - Wesley Ivan Hurt, Sep 10 2013
Let r(n) = (a(n)-1)/(a(n)+1) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1) otherwise; then Product_{n>=1} r(n) = (9/7) * (28/26) * (124/126) * (344/342) * (1332/1330) * ... = 48/35. - Dimitris Valianatos, Mar 06 2020
There exist 5 groups of order p^3, when p prime, so this is a subsequence of A054397. Three of them are abelian: C_p^3, C_p^2 X C_p and C_p X C_p X C_p = (C_p)^3. For 8 = 2^3, the 2 nonabelian groups are D_8 and Q_8; for odd prime p, the 2 nonabelian groups are (C_p x C_p) : C_p, and C_p^2 : C_p (remark, for p = 2, these two semi-direct products are isomorphic to D_8). Here C, D, Q mean Cyclic, Dihedral, Quaternion groups of the stated order; the symbols X and : mean direct and semidirect products respectively. - Bernard Schott, Dec 11 2021

			a(3) = 125; since the 3rd prime is 5, a(3) = 5^3 = 125.

Edmund Landau, Elementary Number Theory, translation by Jacob E. Goodman of Elementare Zahlentheorie (Vol. I_1 (1927) of Vorlesungen über Zahlentheorie), by Edmund Landau, with added exercises by Paul T. Bateman and E. E. Kohlbecker, Chelsea Publishing Co., New York, 1958, pp. 31-32.

T. D. Noe, Table of n, a(n) for n = 1..1000
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Eric Weisstein's World of Mathematics, Prime Power.
Wikipedia, p-group, Classification.
Index to sequences related to prime signature

Other sequences that are k-th powers of primes are: A000040 (k=1), A001248 (k=2), this sequence (k=3), A030514 (k=4), A050997 (k=5), A030516 (k=6), A092759 (k=7), A179645 (k=8), A179665 (k=9), A030629 (k=10), A079395 (k=11), A030631 (k=12), A138031 (k=13), A030635 (k=16), A138032 (k=17), A030637 (k=18).
Cf. A060800, A131991, A000578, subsequence of A046099.
Subsequence of A007422 and of A054397.
Cf. A036966, A085541, A088453, A157289, A258600.

a030078 = a000578 . a000040
a030078_list = map a000578 a000040_list -- Reinhard Zumkeller, May 26 2012

[p^3: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

Array[Prime[ # ]^3&, 5! ] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

a(n)=prime(n)^3 \\ Charles R Greathouse IV, Mar 20 2013

from sympy import prime, primerange
def aupton(terms): return [p**3 for p in primerange(1, prime(terms)+1)]
print(aupton(35)) # Michael S. Branicky, Aug 27 2021

[p**3 for p in prime_range(100)] # Zerinvary Lajos, May 15 2007

n such that A062799(n) = 3. - Benoit Cloitre, Apr 06 2002
a(n) = A000040(n)^3. - Omar E. Pol, Jul 27 2009
A064380(a(n)) = A000010(a(n)). - Vladimir Shevelev, Apr 19 2010
A003415(a(n)) = A079705(n). - Reinhard Zumkeller, Jun 26 2011
A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011
A000005(a(n)) = 4. - Wesley Ivan Hurt, Sep 10 2013
a(n) = A119959(n) * A008864(n) -1.- R. J. Mathar, Aug 13 2019
Sum_{n>=1} 1/a(n) = P(3) = 0.1747626392... (A085541). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(3)/zeta(6) (A157289).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(3) (A088453). (End)

A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336

Offset: 1

Eric W. Weisstein

nonn

Also called cubeful numbers, but this term is ambiguous and is best avoided.
Numbers n such that A007427(n) = sum(d|n,mu(d)*mu(n/d)) == 0. - Benoit Cloitre, Apr 17 2002
The convention in the OEIS is that squareful, cubeful, biquadrateful (A046101), ... mean the same as "not squarefree" etc., while 2- or square-full, 3- or cube-full (A036966), 4-full (A036967) are used for Golomb's notion of powerful numbers (A001694, see references there), when each prime factor occurs to a power > 1. - M. F. Hasler, Feb 12 2008. Added by N. J. A. Sloane, Apr 25 2023:  This suggestion has not been a success. It is hopeless to try to make a distinction between "cubeful" and "cubefull". To avoid ambiguity, do not use either term, but instead say exactly what you mean.
Also solutions to equation tau_{-2}(n)=0, where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013
The asymptotic density of this sequence is 1 - 1/zeta(3) = 0.168092... - Amiram Eldar, Jul 09 2020

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cubefree.

Complement of A004709.
Subsequences: A000578 and A030078.
Cf. A001694, A036966, A046101, A088453.

a046099 n = a046099_list !! (n-1)
a046099_list = filter ((== 1) . a212793) [1..]
-- Reinhard Zumkeller, May 27 2012

isA046099 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) >= 3 then
            return true;
        end if;
    end do:
    false ;
end proc:
for n from 1 do
    if isA046099(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, Dec 08 2015

lst={};Do[a=0;Do[If[FactorInteger[m][[n, 2]]>2, a=1], {n, Length[FactorInteger[m]]}];If[a==1, AppendTo[lst, m]], {m, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 15 2008 *)

is(n)=n>7 && vecmax(factor(n)[,2])>2 \\ Charles R Greathouse IV, Sep 17 2015

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 3) != n
print(list(filter(ok, range(1, 337)))) # Michael S. Branicky, Aug 16 2021

from sympy import mobius, integer_nthroot
def A046099(n):
    def f(x): return n+sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

A212793(a(n)) = 0. - Reinhard Zumkeller, May 27 2012

Sum_{n>=1} 1/a(n)^s = (zeta(s)*(zeta(3*s)-1))/zeta(3*s). - Amiram Eldar, Dec 27 2022

More terms from Vladimir Joseph Stephan Orlovsky, Aug 15 2008

Edited by N. J. A. Sloane, Jul 27 2009

A076467 Perfect powers m^k where m is a positive integer and k >= 3.

Original entry on oeis.org

1, 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

Robert G. Wilson v, Oct 14 2002

nonn

If p|n with p prime then p^3|n.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 1..250 from Reinhard Zumkeller)

Subsequence of A036966.

Cf. A001597, A076468, A076469, A076470.

a076467 n = a076467_list !! (n-1)
a076467_list = 1 : filter ((> 2) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

import qualified Data.Set as Set (null)
import Data.Set (empty, insert, deleteFindMin)
a076467 n = a076467_list !! (n-1)
a076467_list = 1 : f [2..] empty where
   f xs'@(x:xs) s | Set.null s || m > x ^ 3 = f xs $ insert (x ^ 3, x) s
                  | m == x ^ 3  = f xs s
                  | otherwise = m : f xs' (insert (m * b, b) s')
                  where ((m, b), s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 18 2013

N:= 10^5: # to get all terms <= N
S:= {1, seq(seq(m^k, m = 2 .. floor(N^(1/k))),k=3..ilog2(N))}:
sort(convert(S,list)); # Robert Israel, Sep 30 2015

a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 2, a = Append[a, n]; Print[n]], {n, 2, 17575}]; a
(* Second program: *)
n = 10^5; Join[{1}, Table[m^k, {k, 3, Floor[Log[2, n]]}, {m, 2, Floor[n^(1/k)]}] // Flatten // Union] (* Jean-François Alcover, Feb 13 2018, after Robert Israel *)

is(n)=ispower(n)>2||n==1 \\ Charles R Greathouse IV, Sep 03 2015, edited for n=1 by M. F. Hasler, May 26 2018

A076467(lim)={my(L=List(1),lim2=logint(lim,2),m,k);for(k=3,lim2, for(m=2,sqrtnint(lim,k),listput(L, m^k);));listsort(L,1);L}
b076467(lim)={my(L=A076467(lim)); for(i=1,#L,print(i ," ",L[i]));} \\ Anatoly E. Voevudko, Sep 29 2015, edited by M. F. Hasler, May 25 2018

A076467_vec(LIM,S=List(1))={for(x=2,sqrtnint(LIM,3),for(k=3, logint(LIM, x), listput(S, x^k))); Set(S)} \\ M. F. Hasler, May 25 2018

from sympy import mobius, integer_nthroot
def A076467(n):
    def f(x): return int(n-1+x-integer_nthroot(x,4)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]-2) for k in range(3,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

For n > 1: GCD(exponents in prime factorization of a(n)) > 2, cf. A124010. - Reinhard Zumkeller, Apr 13 2012

Sum_{n>=1} 1/a(n) = 2 - zeta(2) + Sum_{k>=2} mu(k)*(2 - zeta(k) - zeta(2*k)) = 1.3300056287... - Amiram Eldar, Jul 02 2022

Edited by Robert Israel, Sep 30 2015

A023052 Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, 54748, 92727, 93084, 194979, 548834, 1741725, 4210818, 9800817, 9926315, 14459929, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153

Offset: 1

David W. Wilson

The old name was "Powerful numbers, definition (3)". Cf. A001694, A007532. - N. J. A. Sloane, Jan 16 2022.
Randle has suggested that these numbers be called "powerful", but this usually refers to a distinct property related to prime factorization, cf. A001694, A036966, A005934.
Numbers m such that m = Sum_{i=1..k} d(i)^s for some s, where d(1..k) are the decimal digits of m.
Superset of A005188 (Plusperfect, narcissistic or Armstrong numbers: s=k), A046197 (s=3), A052455 (s=4), A052464 (s=5), A124068 (s=6, 7), A124069 (s=8). - R. J. Mathar, Jun 15 2009, Jun 22 2009

			153 = 1^3 + 5^3 + 3^3, 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.

Jerome Raulin, Table of n, a(n) for n = 1..345 (terms 1..255 from Joseph Myers)
Encyclopaedia Britannica, Perfect digital invariant, article "Number patterns and curiosities" online since July 26, 1999, revised Aug 25, 2000.
Hans Havermann, Extended table of values for A023052 and A046074
Donald E. Knuth, The Art of Computer Programming, Volume 4, Pre-Fascicle 9B A Potpourri of Puzzles
J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383.
J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Narcissistic Number
Index entries for sequences related to powerful numbers

Cf. A001694 (powerful numbers: p|n => p^2|n), A005934 (highly powerful numbers).
Cf. A003321, A007532, A014576, A046074, A046761, A053540, A161752.
Cf. A005188 (here the power must be equal to the number of digits).
In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9).

Select[Range[0, 10^5], Function[m, AnyTrue[Function[k, Total@ Map[Power[#, k] &, IntegerDigits@ m]] /@ Range@ 10, # == m &]]] (* Michael De Vlieger, Feb 08 2016, Version 10 *)

is(n)=if(n<10, return(1)); my(d=digits(n),m=vecmax(d)); if(m<2, return(0)); for(k=3,logint(n,m), if(sum(i=1,#d,d[i]^k)==n, return(1))); 0 \\ Charles R Greathouse IV, Feb 06 2017

select( is_A023052(n,b=10)={nn|| return(t==n))}, [0..10^5]) \\  M. F. Hasler, Nov 21 2019

Computed to 10^50 by G. N. Gusev (GGN(AT)rm.yaroslavl.ru)
Computed to 10^74 by Xiaoqing Tang
A-number typo corrected by R. J. Mathar, Jun 22 2009
Computed to 10^105 by Joseph Myers
Cross-references edited by Joseph Myers, Jun 28 2009
Edited by M. F. Hasler, Nov 21 2019

cp's OEIS Frontend

A363177 Primitive abundant numbers (A071395) that are cubefull numbers (A036966).

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A210470 Powerful numbers (A001694) which can be written as the sum of two relatively prime 3-powerful numbers (A036966) different from 1.

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A362972 Squarefree kernels of cubefull numbers (A036966).

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A363013 a(n) is the number of prime factors (counted with multiplicity) of the n-th cubefull number (A036966).

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A363014 Cubefull numbers (A036966) with a record gap to the next cubefull number.

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Mathematica

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A307703 Highly powerful numbers (A005934) that are not cubeful (A036966).

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Mathematica

A030078 Cubes of primes.

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Haskell

Magma

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A046099 Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709.