A118896 -id:A118896 - cp's OEIS frontend

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000

Offset: 1

N. J. A. Sloane

Numbers of the form a^2*b^3, a >= 1, b >= 1.
In other words, if the prime factorization of n is Product_k p_k^e_k then all e_k are greater than 1.
Numbers n such that Sum_{d|n} phi(d)*phi(n/d)*mu(d) > 0; places of nonzero A300717. - Benoit Cloitre, Nov 30 2002
This sequence is closed under multiplication. The primitive elements are A168363. - Franklin T. Adams-Watters, May 30 2011
Complement of A052485. - Reinhard Zumkeller, Sep 16 2011
The number of terms less than or equal to 10^k beginning with k = 0: 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, ...: A118896. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, ... . - Robert G. Wilson v, Aug 15 2014
a(m) mod prime(n) > 0 for m < A258599(n); a(A258599(n)) = A001248(n) = prime(n)^2. - Reinhard Zumkeller, Jun 06 2015
From Des MacHale, Mar 07 2021: (Start)
A number m is powerful if and only if |R/Z(R)| = m, for some finite non-commutative ring R.
A number m is powerful if and only if |G/Z(G)| = m, for some finite nilpotent class two group G (Reference Aine Nishe). (End)
Numbers n such that Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)) > 0. - Richard L. Ollerton, May 09 2021

			1 is a term because for every prime p that divides 1, p^2 also divides 1.
2 is not a term since 2 divides 2 but 2^2 does not.
4 is a term because 2 is the only prime that divides 4 and 2^2 does divide 4. - _N. J. A. Sloane_, Jan 16 2022

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.
Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
Richard A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
Aine NiShe, Commutativity and Generalisations in Finite Groups, Ph.D. Thesis, University College Cork, 2000.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 9.1 Potente Zahlen, pp. 241-247.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 54, exercise 10 (in the third edition 2015, p. 63, exercise 70).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..5000 from G. C. Greubel)
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Valentin Blomer, Binary quadratic forms with large discriminants and sums of two squareful numbers II, Journal of the London Mathematical Society 71:1 (2005), pp. 69-84.
Chris K. Caldwell, Powerful Numbers.
Tsz Ho Chan, Arithmetic Progressions Among Powerful Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.1.
Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv:2503.21485 [math.NT], 2025.
Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Powerful Values of Quadratic Polynomials, J. Int. Seq. 14 (2011), Article 11.3.3.
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102. [Zahlen i-ter Art, p. 101]
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
K. Schneider, PlanetMath.org, Squarefull Number.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, The distribution of square-full integers, Ark. Mat., Volume 11, Number 1-2 (1973), 195-201.
Eric Weisstein's World of Mathematics, Powerful Number.
Eric Weisstein's World of Mathematics, Squareful.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Disjoint union of A062503 and A320966.
Cf. A007532 (Powerful numbers, definition (2)), A005934, A005188, A003321, A014576, A023052 (Powerful numbers, definition (3)), A046074, A013929, A076871, A258599, A001248, A112526, A168363, A224866, A261883, A300717.
Cf. A052485 (complement), A076446 (first differences), A376361, A376362.

a001694 n = a001694_list !! (n-1)
a001694_list = filter ((== 1) . a112526) [1..]
-- Reinhard Zumkeller, Nov 30 2012

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A001694 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001694(a) then return a; end if; end do; end if; end proc:
seq(A001694(n),n=1..20) ; # R. J. Mathar, Jun 07 2011

Join[{1}, Select[ Range@ 1250, Min@ FactorInteger[#][[All, 2]] > 1 &]]
(* Harvey P. Dale, Sep 18 2011; modified by Robert G. Wilson v, Aug 11 2014 *)
max = 10^3; Union@ Flatten@ Table[a^2*b^3, {b, max^(1/3)}, {a, Sqrt[max/b^3]}] (* Robert G. Wilson v, Aug 11 2014 *)
nextPowerfulNumber[n_] := Block[{r = Range[ Floor[1 + n^(1/3)]]^3}, Min@ Select[ Sort[ r*Floor[1 + Sqrt[n/r]]^2], # > n &]]; NestList[ nextPowerfulNumber, 1, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isA001694(n)=n=factor(n)[,2];for(i=1,#n,if(n[i]==1,return(0)));1 \\ Charles R Greathouse IV, Feb 11 2011

list(lim,mn=2)=my(v=List(),t); for(m=1,sqrtnint(lim\1,3), t=m^3; for(n=1,sqrtint(lim\t), listput(v,t*n^2))); Set(v) \\ Charles R Greathouse IV, Jul 31 2011; edited Sep 22 2015

is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012

from sympy import factorint
A001694 = [1]+[n for n in range(2,10**6) if min(factorint(n).values()) > 1]
# Chai Wah Wu, Aug 14 2014

from math import isqrt
from sympy import mobius, integer_nthroot
def A001694(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

sloane.A001694.list(54) # Peter Luschny, Feb 08 2015

A112526(a(n)) = 1. - Reinhard Zumkeller, Sep 16 2011
Bateman & Grosswald prove that there are zeta(3/2)/zeta(3) x^{1/2} + zeta(2/3)/zeta(2) x^{1/3} + O(x^{1/6}) terms up to x; see section 5 for a more precise error term. - Charles R Greathouse IV, Nov 19 2012
a(n) = A224866(n) - 1. - Reinhard Zumkeller, Jul 23 2013
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6). - Ivan Neretin, Aug 30 2015
Sum_{n>=1} 1/a(n)^s = zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1/2 (Golomb, 1970). - Amiram Eldar, Oct 02 2022

More terms from Henry Bottomley, Mar 16 2000

Definition expanded by Jonathan Sondow, Jan 03 2016

A052486 Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power.

Original entry on oeis.org

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6075

Offset: 1

Henry Bottomley, Mar 16 2000

nonn

Number of terms < 10^n: 0, 1, 13, 60, 252, 916, 3158, 10553, 34561, 111891, 359340, 1148195, 3656246, 11616582, 36851965, ..., A118896(n) - A070428(n). - Robert G. Wilson v, Aug 11 2014

a(n) = (s(n))^2 * f(n), s(n) > 1, f(n) > 1, where s(n) is not a power of f(n), and f(n) is squarefree and gcd(s(n), f(n)) = f(n). - Daniel Forgues, Aug 11 2015

			a(3)=200 because 200=2^3*5^2, both 3 and 2 are greater than 1, and the highest common factor of 3 and 2 is 1.
Factorizations of a(1) to a(20):
    72 = 2^3  3^2,  108 = 2^2 3^3,  200 = 2^3 5^2,  288 = 2^5  3^2,
   392 = 2^3  7^2,  432 = 2^4 3^3,  500 = 2^2 5^3,  648 = 2^3  3^4,
   675 = 3^3  5^2,  800 = 2^5 5^2,  864 = 2^5 3^3,  968 = 2^3 11^2,
   972 = 2^2  3^5, 1125 = 3^2 5^3, 1152 = 2^7 3^2, 1323 = 3^3  7^2,
  1352 = 2^3 13^2, 1372 = 2^2 7^3, 1568 = 2^5 7^2, 1800 = 2^3  3^2 5^2.
Examples for a(n) = (s(n))^2 * f(n): (see above comment)
s(n) = 6,  6, 10, 12, 14, 12, 10, 18, 15, 20, 12, 22, 18, 15, 24, 21,
f(n) = 2,  3,  2,  2,  2,  3,  5,  2,  3,  2,  6,  2,  3,  5,  2,  3,

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Project Euler, Problem 302: Strong Achilles Numbers.
Robert Israel, log-log plot of a(n).
Eric Weisstein's World of Mathematics, Achilles Number.
OEIS Wiki, Achilles numbers.

Cf. A001597, A001694, A007916, A072102, A082695.

filter:= proc(n) local E; E:= map(t->t[2], ifactors(n)[2]); min(E)>1 and igcd(op(E))=1 end proc:
select(filter,[$1..10000]); # Robert Israel, Aug 11 2014

achillesQ[n_] := Block[{ls = Last /@ FactorInteger@n}, Min@ ls > 1 == GCD @@ ls]; Select[ Range@ 5500, achillesQ@# &] (* Robert G. Wilson v, Jun 10 2010 *)

is(n)=my(f=factor(n)[,2]); n>9 && vecmin(f)>1 && gcd(f)==1 \\ Charles R Greathouse IV, Sep 18 2015, replacing code by M. F. Hasler, Sep 23 2010

from math import gcd
from itertools import count, islice
from sympy import factorint
def A052486_gen(startvalue=1): # generator of terms >= startvalue
    return (n for n in count(max(startvalue,1)) if (lambda x: all(e > 1 for e in x) and gcd(*x) == 1)(factorint(n).values()))
A052486_list = list(islice(A052486_gen(),20)) # Chai Wah Wu, Feb 19 2022

from math import isqrt
from sympy import mobius, integer_nthroot
def A052486(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x+1, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length()))
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

a(n) = O(n^2). - Daniel Forgues, Aug 11 2015
a(n) = O(n^2 / log log n). - Daniel Forgues, Aug 12 2015
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} mu(k)*(1-zeta(k)) - 1 = A082695 - A072102 - 1 = 0.06913206841581433836...  - Amiram Eldar, Oct 14 2020

Example edited by Mac Coombe (mac.coombe(AT)gmail.com), Sep 18 2010

Name edited by M. F. Hasler, Jul 17 2019

A362973 The number of cubefull numbers (A036966) not exceeding 10^n.

Original entry on oeis.org

1, 2, 7, 20, 51, 129, 307, 713, 1645, 3721, 8348, 18589, 41136, 90619, 198767, 434572, 947753, 2062437, 4480253, 9718457, 21055958, 45575049, 98566055, 213028539, 460160083, 993533517, 2144335391, 4626664451, 9980028172, 21523027285, 46408635232, 100053270534

Offset: 0

Amiram Eldar, May 11 2023

nonn

The number of cubefull numbers not exceeding x is N(x) = c_0 * x^(1/3) + c_1 * x^(1/4) + c_2 * x^(1/5) + o(x^(1/8)), where c_0 (A362974), c_1 (A362975) and c_2 (A362976) are constants (Bateman and Grosswald, 1958; Finch, 2003).

The digits of a(3k) converge to A362974 as k -> oo. - Chai Wah Wu, May 13 2023

			There are 2 cubefull numbers not exceeding 10, 1 and 8, therefore a(1) = 2.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.6.1, pp. 113-115.

Chai Wah Wu, Table of n, a(n) for n = 0..36
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois Journal of Mathematics, Vol. 2, No. 1 (1958), pp. 88-98.
A. Ivić and P. Shiu, The distribution of powerful integers, Illinois Journal of Mathematics, Vol. 26, No. 4 (1982), pp. 576-590.
Ekkehard Krätzel, On the distribution of square-full and cube-full numbers, Monatshefte für Mathematik, Vol. 120, No. 2 (1995), pp. 105-119.
P. Shiu, The distribution of cube-full numbers, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295.
P. Shiu, Cube-full numbers in short intervals, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 112, No. 1 (1992), pp. 1-5.

Cf. A036966, A362974, A362975, A362976.

Similar sequences: A070428, A118896.

a[n_] := Module[{max = 10^n}, CountDistinct@ Flatten@ Table[i^5 * j^4 * k^3, {i, Surd[max, 5]}, {j, Surd[max/i^5, 4]}, {k, CubeRoot[max/(i^5*j^4)]}]]; Array[a, 15, 0]

from math import gcd
from sympy import factorint, integer_nthroot
def A362973(n):
    m, c = 10**n, 0
    for x in range(1,integer_nthroot(m,5)[0]+1):
        if all(d<=1 for d in factorint(x).values()):
            for y in range(1,integer_nthroot(z:=m//x**5,4)[0]+1):
                if gcd(x,y)==1 and all(d<=1 for d in factorint(y).values()):
                    c += integer_nthroot(z//y**4,3)[0]
    return c # Chai Wah Wu, May 11-13 2023

a(23)-a(31) from Chai Wah Wu, May 11 2023

A376092 10^n-th powerful number.

Original entry on oeis.org

1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, 212104218976916644, 21190268970925690248, 2118092209873957381248, 211765852717674823741924, 21174572668805230623003225, 2117363857447354911021280900

Offset: 0

Chai Wah Wu, Sep 09 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..16

Cf. A001694, A118896.

from math import isqrt
from sympy import mobius, integer_nthroot
def A376092(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    m = 10**n
    def f(x):
        c, l = m+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return bisection(f,m,m)

a(n) = A001694(10^n).

Limit_{n->oo} a(n)/10^(2n) = (zeta(3)/zeta(3/2))^2 = 0.21172829478335...

A380254 Number of powerful numbers (in A001694) that do not exceed primorial A002110(n).

Original entry on oeis.org

1, 1, 2, 7, 22, 85, 330, 1433, 6450, 31555, 172023, 964560, 5891154, 37807505, 248226019, 1702890101, 12401685616, 95277158949, 744210074157, 6091922351106, 51332717836692, 438592279944173, 3898316990125822, 35515462315592564, 335052677538616216, 3299888425002527366

Offset: 0

Michael De Vlieger, Jan 19 2025

In other words, A001694(a(n)) is the largest powerful number less than or equal to A002110(n).

			Let P = A002110 and let s = A001694.
a(0) = 1 since P(0) = 1, and the set s(1) = {1} contains k that do not exceed 1.
a(1) = 1 since P(1) = 2, and the set s(1) = {1} contains k <= 2.
a(2) = 2 since P(2) = 6, and the set s(1..2) = {1, 4} contains k <= 6.
a(3) = 7 since P(3) = 30, and the set s(1..7) = {1, 4, 8, 9, 16, 25, 27} contains k <= 30.
a(4) = 22 since P(4) = 210, and the set s(1..19) = {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200} contains k <= 210, etc.

Chai Wah Wu, Table of n, a(n) for n = 0..26

Cf. A001694, A002110, A062762, A118896, A380337.

f[x_] := Sum[If[SquareFreeQ[ii], Floor[Sqrt[x/ii^3]], 0], {ii, x^(1/3)}];
Table[f[#[[k + 1]]], {k, 0, Length[#] - 1}] &[
  FoldList[Times, 1, Prime[Range[12] ] ] ] (* function f after Robert G. Wilson v at A118896 *)

from math import isqrt
from sympy import primorial, integer_nthroot, mobius
def A380254(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    if n == 0: return 1
    m = primorial(n)
    c, l, j = squarefreepi(integer_nthroot(m, 3)[0]), 0, isqrt(m)
    while j>1:
        k2 = integer_nthroot(m//j**2,3)[0]+1
        w = squarefreepi(k2-1)
        c += j*(w-l)
        l, j = w, isqrt(m//k2**3)
    return c-l # Chai Wah Wu, Jan 24 2025

a(18)-a(25) from Chai Wah Wu, Jan 24 2025

A363170 The number of powerful abundant numbers (A363169) not exceeding 10^n.

Original entry on oeis.org

0, 3, 23, 82, 297, 1000, 3268, 10534, 33799, 107901, 343155, 1090189, 3460380, 10970774, 34749182, 109991778, 348006756, 1101058505, 3483105232, 11017518803

Offset: 1

Amiram Eldar, May 19 2023

The ratios a(n)/A118896(n) seem to converge to a positive value as n grows: for n = 14..20 they are 0.506417..., 0.506728..., 0.506863..., 0.506890..., 0.506987..., 0.507059..., 0.507120... .

Conjecture: the asymptotic relative density of the abundant numbers within the powerful numbers exists and equals 0.507... .

			a(2) = 3 since there are 3 powerful abundant numbers not exceeding 10^2: 36, 72 and 100.

Cf. A001694, A005101, A118896, A302992, A363169.

seq[nmax_] := Module[{c = 0, p = 10, k = 1, kmax = 10^nmax, s = {}}, While[k <= kmax, If[DivisorSigma[-1, k] > 2 && Min[FactorInteger[k][[;;, 2]]] > 1, c++]; If[k == p, AppendTo[s, c]; p *= 10]; k++]; s]; seq[5]

is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; } \\ A363169
lista(nmax) = {my(c = 0, p = 10, k = 1, kmax = 10^nmax); while(k <= kmax, if(is(k), c++); if(k == p, print1(c, ", "); p *= 10); k++); }

A381496 Number of powerful numbers that are not prime powers that do not exceed 10^n.

Original entry on oeis.org

0, 0, 3, 28, 133, 510, 1790, 5997, 19639, 63541, 204037, 652173, 2078320, 6609816, 20993381, 66612867, 211222374, 669428537, 2120835892, 6717184256, 21270247404, 67341572823, 213173925948, 674739560651, 2135491756895, 6758117426102, 21385762133815, 67670426242420

Offset: 0

Michael De Vlieger, Feb 25 2025

nonn

Number of k such that omega(k) > 1 and rad(k)^2 | k (i.e., in A286708) that do not exceed 10^n, where omega = A001221 and rad = A007947.

			Let S = A286708 = A001694 \ A246547 = A126706 \ A001694.
a(0) = a(1) = 0 since 36 is the smallest term in S.
a(2) = 3 since S(1..3) = {36, 72, 100}.
a(3) = 28 since S(4..28) = {108, 144, ..., 972, 1000}.
a(4) = 133 since S(29..133) = {1089, 1125, ..., 9801, 10000}, etc.

Cf. A001694, A118896, A126706, A246547, A267574, A286708, A380430, A380431, A381391.

Table[Sum[Boole[SquareFreeQ[k]]*Floor[Sqrt[10^n/k^3]], {k, 10^(n/3)}] - Sum[PrimePi[10^(n/k)], {k, 2, n*Log2[10]}] - 1, {n, 0, 12}]

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A381496(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    m, l = 10**n, 0
    j, c = isqrt(m), -1-sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length())),
    while j>1:
        k2 = integer_nthroot(m//j**2,3)[0]+1
        w = squarefreepi(k2-1)
        c += j*(w-l)
        l, j = w, isqrt(m//k2**3)
    return c+squarefreepi(integer_nthroot(m,3)[0])-l # Chai Wah Wu, Feb 25 2025

a(n) = -1 + Sum_{k=1..10^(n/3)} [rad(k)=k]*floor(sqrt(10^n/k^3)) - Sum_{k=2..n*log_2(10)} pi(10^(n/k)).
a(n) = -1 + A118896(n) - A267574(n).
a(n) < A381391(n) for n > 0 since A286708 is a proper subset of A126706.

cp's OEIS Frontend

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

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A052486 Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power.

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A362973 The number of cubefull numbers (A036966) not exceeding 10^n.

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A376092 10^n-th powerful number.

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A380254 Number of powerful numbers (in A001694) that do not exceed primorial A002110(n).

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A363170 The number of powerful abundant numbers (A363169) not exceeding 10^n.

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