A090699 -id:A090699 - cp's OEIS frontend

A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 6, 1, 26, 3, 7, 29, 30, 31, 2, 33, 34, 35, 1, 37, 38, 39, 10, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 6, 55, 14, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2, 73, 74, 3, 19, 77

Offset: 1

R. Muller, Mar 15 1996

Also called core(n). [Not to be confused with the squarefree kernel of n, A007947.]
Sequence read mod 4 gives A065882. - Philippe Deléham, Mar 28 2004
This is an arithmetic function and is undefined if n <= 0.
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), lcm(A007947(b),c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n. [Corrected by M. F. Hasler, Mar 01 2018]
If n > 1, the quantity f(n) = log(n/core(n))/log(n) satisfies 0 <= f(n) <= 1; f(n) = 0 when n is squarefree and f(n) = 1 when n is a perfect square. One can define n as being "epsilon-almost squarefree" if f(n) < epsilon. - Kurt Foster (drsardonicus(AT)earthlink.net), Jun 28 2008
a(n) is the smallest natural number m such that product of geometric mean of the divisors of n and geometric mean of the divisors of m are integers. Geometric mean of the divisors of number n is real number b(n) = Sqrt(n). a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A000290: a(A000290(n)) = 1. For n = 8; b(8) = sqrt(8), a(n) = 2 because b(2) = sqrt(2); sqrt(8) * sqrt(2) = 4 (integer). - Jaroslav Krizek, Apr 26 2010
Dirichlet convolution of A010052 with the sequence of absolute values of A055615. - R. J. Mathar, Feb 11 2011
Booker, Hiary, & Keating outline a method for bounding (on the GRH) a(n) for large n using L-functions. - Charles R Greathouse IV, Feb 01 2013
According to the formula a(n) = n/A000188(n)^2, the scatterplot exhibits the straight lines y=x, y=x/4, y=x/9, ..., i.e., y=x/k^2 for all k=1,2,3,... - M. F. Hasler, May 08 2014
The Dirichlet inverse of this sequence is A008836(n) * A063659(n). - Álvar Ibeas, Mar 19 2015
a(n) = 1 if n is a square, a(n) = n if n is a product of distinct primes. - Zak Seidov, Jan 30 2016
All solutions of the Diophantine equation n*x=y^2 or, equivalently, G(n,x)=y, with G being the geometric mean, are of the form x=k^2*a(n), y=k*sqrt(n*a(n)), where k is a positive integer. - Stanislav Sykora, Feb 03 2016
If f is a multiplicative function then Sum_{d divides n} f(a(d)) is also multiplicative. For example, A010052(n) = Sum_{d divides n} mu(a(d)) and A046951(n) = Sum_{d divides n} mu(a(d)^2). - Peter Bala, Jan 24 2024

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)
Krassimir T. Atanassov, On Some of Smarandache's Problems.
Krassimir T. Atanassov, On the 22nd, 23rd, and the 24th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 2, 80-82.
Andrew Booker, Ghaith Hiary, and Jon Keating, Detecting squarefree numbers, CNTA XII (2012).
Henry Bottomley, Some Smarandache-type multiplicative sequences.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Vlad Copil and Laurenţiu Panaitopol, Properties of a sequence generated by positive integers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 2 (2007), pp. 131-137; alternative link.
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Eric Weisstein's World of Mathematics, Squarefree Part.

See A000188, A007947, A008833, A019554, A117811 for related information, specific to n.
See A027746, A027748, A124010 for factorization data for n.
Analogous sequences: A050985, A053165, A055231.
Cf. A002734, A005117 (range of values), A059897, A069891 (partial sums), A090699, A350389.
Related to A006519 via A225546.

a007913 n = product $
            zipWith (^) (a027748_row n) (map (`mod` 2) $ a124010_row n)
-- Reinhard Zumkeller, Jul 06 2012

[ Squarefree(n) : n in [1..256] ]; // N. J. A. Sloane, Dec 23 2006

A007913 := proc(n) local f,a,d; f := ifactors(n)[2] ; a := 1 ; for d in f do if type(op(2,d),'odd') then a := a*op(1,d) ; end if; end do: a; end proc: # R. J. Mathar, Mar 18 2011
# second Maple program:
a:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 20 2015
seq(n / expand(numtheory:-nthpow(n, 2)), n=1..77);  # Peter Luschny, Jul 12 2022

data = Table[Sqrt[n], {n, 1, 100}]; sp = data /. Sqrt[] -> 1; sfp = data/sp /. Sqrt[x] -> x (* Artur Jasinski, Nov 03 2008 *)
Table[Times@@Power@@@({#[[1]],Mod[ #[[2]],2]}&/@FactorInteger[n]),{n,100}] (* Zak Seidov, Apr 08 2009 *)
Table[{p, e} = Transpose[FactorInteger[n]]; Times @@ (p^Mod[e, 2]), {n, 100}] (* T. D. Noe, May 20 2013 *)
Sqrt[#] /. (c_:1)*a_^(b_:0) -> (c*a^b)^2& /@ Range@100 (* Bill Gosper, Jul 18 2015 *)

PARI
```
a(n)=core(n)
```

from sympy import factorint, prod
def A007913(n):
    return prod(p for p, e in factorint(n).items() if e % 2)
# Chai Wah Wu, Feb 03 2015

[squarefree_part(n) for n in (1..77)] # Peter Luschny, Feb 04 2015

Multiplicative with a(p^k) = p^(k mod 2). - David W. Wilson, Aug 01 2001
a(n) modulo 2 = A035263(n); a(A036554(n)) is even; a(A003159(n)) is odd. - Philippe Deléham, Mar 28 2004
Dirichlet g.f.: zeta(2s)*zeta(s-1)/zeta(2s-2). - R. J. Mathar, Feb 11 2011
a(n) = n/( Sum_{k=1..n} floor(k^2/n)-floor((k^2 -1)/n) )^2. - Anthony Browne, Jun 06 2016
a(n) = rad(n)/a(n/rad(n)), where rad = A007947. This recurrence relation together with a(1) = 1 generate the sequence. - Velin Yanev, Sep 19 2017
From Peter Munn, Nov 18 2019: (Start)
a(k*m) = A059897(a(k), a(m)).
a(n) = n / A008833(n).
(End)
a(A225546(n)) = A225546(A006519(n)). - Peter Munn, Jan 04 2020
From Amiram Eldar, Mar 14 2021: (Start)
Theorems proven by Copil and Panaitopol (2007):
Lim sup_{n->oo} a(n+1)-a(n) = oo.
Lim inf_{n->oo} a(n+1)-a(n) = -oo.
Sum_{k=1..n} 1/a(k) ~ c*sqrt(n) + O(log(n)), where c = zeta(3/2)/zeta(3) (A090699). (End)
a(n) = A019554(n)^2/n. - Jianing Song, May 08 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/30 = 0.328986... . - Amiram Eldar, Oct 25 2022
a(n) = A007947(A350389(n)). - Amiram Eldar, Jan 20 2024

More terms from Michael Somos, Nov 24 2001

Definition reformulated by Daniel Forgues, Mar 24 2009

A362974 Decimal expansion of Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)).

Original entry on oeis.org

4, 6, 5, 9, 2, 6, 6, 1, 2, 2, 5, 0, 0, 6, 5, 6, 9, 4, 1, 2, 7, 7, 4, 3, 1, 1, 0, 8, 9, 1, 3, 6, 2, 5, 8, 6, 2, 1, 3, 0, 5, 4, 3, 3, 6, 7, 2, 8, 3, 2, 5, 6, 5, 3, 8, 4, 7, 5, 7, 6, 9, 2, 4, 0, 1, 5, 3, 0, 3, 4, 1, 8, 0, 8, 6, 5, 7, 3, 5, 2, 3, 8, 7, 2, 1, 8, 0, 7, 7, 5, 8, 9, 0, 2, 6, 8, 4, 6, 2, 3, 4, 9, 0, 9, 7

Offset: 1

Amiram Eldar, May 11 2023

The coefficient c_0 of the leading term in the asymptotic formula for the number of cubefull numbers (A036966) not exceeding x, N(x) = c_0 * x^(1/3) + c_1 * x^(1/4) + c_2 * x^(1/5) + o(x^(1/8)) (Bateman and Grosswald, 1958; Finch, 2003).

			4.65926612250065694127743110891362586213054336728325...

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

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.
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, A090699 (analogous constant for powerful numbers), A244000, A337736, A362973, A362975 (c_1), A362976 (c_2).

Cf. A051904.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{0, 0, 0, -1, -1}, {0, 0, 0, 4, 5}, m]; RealDigits[(1 + 1/2^(4/3) + 1/2^(5/3)) * (1 + 1/3^(4/3) + 1/3^(5/3)) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/3] - 1/2^(n/3) - 1/3^(n/3))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

PARI
```
prodeulerrat(1 + 1/p^4 + 1/p^5, 1/3)
```

Equals 1 + lim_{m->oo} (1/m) Sum_{k=1..m} A337736(k).

A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

Original entry on oeis.org

1, 4, 8, 9, 25, 27, 36, 49, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 289, 343, 361, 392, 441, 484, 500, 529, 675, 676, 841, 900, 961, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1331, 1352, 1369, 1372, 1444, 1521, 1681, 1764, 1800, 1849, 2116, 2197, 2209

Offset: 1

Amiram Eldar, Oct 22 2020

nonn

Equivalently, numbers k such that if a prime p divides k then p^2 divides k but p^4 does not divide k.
Each term has a unique representation as a^2 * b^3, where a and b are coprime squarefree numbers.
Dehkordi (1998) refers to these numbers as "2-full and 4-free numbers".

			4 = 2^2 is a term since the exponent of its only prime factor is 2.
72 = 2^3 * 3^2 is a terms since the exponents of the primes in its prime factorization are 2 and 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.
Eric Weisstein's World of Mathematics, Biquadratefree.
Eric Weisstein's World of Mathematics, Powerful Number.

Intersection of A001694 and A046100.
Subsequences: A062503, A062838.
Cf. A005117, A090699, A244000, A330595.

Select[Range[2500], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], MemberQ[{2, 3}, #1] &] &]

The number of terms not exceeding x is asymptotically (zeta(3/2)/zeta(3)) * J_2(1/2) * x^(1/2) + (zeta(2/3)/zeta(2)) * J_2(1/3) * x^(1/3), where J_2(s) = Product_{p prime} (1 - p^(-4*s) - p^(-5*s) - p^(-6*s) + p^(-7*s) + p^(-8*s)) (Dehkordi, 1998).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) = 1.748932... (A330595).

A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 2, 0, 0, 3, 1, 0, 1, 0, 1, 1, 0, 2, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 0, 1, 0, 3, 0, 0, 2, 0, 2, 2, 1, 0, 1, 1, 1, 1, 0, 0, 2, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 3, 1, 0, 2, 0, 2, 0, 1, 1, 1, 2, 2, 0

Offset: 1

T. D. Noe, May 09 2006

nonn

Is there an upper bound on the number of powerful numbers between consecutive squares? Pettigrew conjectures that there is no bound. See A119242.

This sequence is unbounded. For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1980). - Amiram Eldar, Jul 10 2020

			a(5) = 2 because the two powerful numbers 27 and 32 are between 25 and 36.

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

Cf. A001694, A090699, A119242.

Powerful[n_] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Table[Length[Select[Range[k^2+1, k^2+2k], Powerful[ # ]&]], {k,130}]

from math import isqrt
from sympy import integer_nthroot, factorint
def A119241(n):
    def f(x): return int(sum(isqrt(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1) if all(d<=1 for d in factorint(k).values())))
    return f((n+1)**2-1)-f(n**2) # Chai Wah Wu, Sep 10 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3/2)/zeta(3) - 1 = A090699 - 1 = 1.173254... - Amiram Eldar, Oct 24 2020

A118896 Number of powerful numbers <= 10^n.

Original entry on oeis.org

1, 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330, 2158391, 6840384, 21663503, 68575557, 217004842, 686552743, 2171766332, 6869227848, 21725636644, 68709456167, 217293374285, 687174291753, 2173105517385, 6872112993377, 21731852479862, 68722847672629, 217322225558934, 687236449779456, 2173239433013146

Offset: 0

Eric W. Weisstein, May 05 2006

nonn

These numbers agree with the asymptotic formula c*sqrt(x), with c=2.1732...(A090699). - T. D. Noe, May 09 2006

Bateman & Grosswald proved that the number of powerful numbers up to x is zeta(3/2)/zeta(3) * x^1/2 + zeta(2/3)/zeta(2) * x^1/3 + o(x^1/6). This approximates the series very closely: up to a(24), all absolute errors are less than 75. - Charles R Greathouse IV, Sep 23 2008

Charles R Greathouse IV and Hiroaki Yamanouchi, Table of n, a(n) for n = 0..45 (terms a(0)-a(32) from Charles R Greathouse IV)
Michael Filaseta and Ognian Trifonov, The distribution of squarefull numbers in short intervals, Acta Arithmetica 67 (1994), pp. 323-333.
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), p. 88-98.
Eric Weisstein's World of Mathematics, Powerful Number

Cf. A001694, A090699.

f:= m -> nops({seq(seq(a^2*b^3, b=1..floor((m/a^2)^(1/3))),a=1..floor(sqrt(m)))}):
seq(f(10^n),n=0..10); # Robert Israel, Aug 12 2014

f[n_] := Block[{max = 10^n}, Length@ Union@ Flatten@ Table[ a^2*b^3, {b, max^(1/3)}, {a, Sqrt[ max/b^3]}]]; Array[f, 13, 0] (* Robert G. Wilson v, Aug 11 2014 *)
powerfulNumberPi[n_] := Sum[ If[ SquareFreeQ@ i, Floor[ Sqrt[ n/i^3]], 0], {i, n^(1/3)}]; Array[ powerfulNumberPi[10^#] &, 27, 0] (* Robert G. Wilson v, Aug 12 2014 *)

a(n)=n=10^n;sum(k=1, floor((n+.5)^(1/3)), if(issquarefree(k), sqrtint(n\k^3))) \\ Charles R Greathouse IV, Sep 23 2008

from math import isqrt
from sympy import integer_nthroot, factorint
def A118896(n):
    m = 10**n
    return sum(isqrt(m//x**3) for x in range(1,integer_nthroot(m,3)[0]+1) if max(factorint(x).values(),default=0)<=1) # Chai Wah Wu, May 13 2023

# faster program
def A118896(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    m, c, l = 10**n, 0, 0
    j = 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)
    c += squarefreepi(integer_nthroot(m,3)[0])-l
    return c # Chai Wah Wu, Sep 09 2024

Pi(x) = Sum_{i=1..x^(1/3)} floor(sqrt(x/i^3)) only if i is squarefree. - Robert G. Wilson v, Aug 12 2014

More terms from T. D. Noe, May 09 2006
a(13)-a(24) from Charles R Greathouse IV, Sep 23 2008
a(25)-a(29) from Charles R Greathouse IV, May 30 2011
a(30) from Charles R Greathouse IV, May 31 2011

A180114 a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).

Original entry on oeis.org

1, 7, 15, 13, 31, 31, 40, 63, 91, 57, 127, 195, 121, 217, 280, 133, 156, 255, 403, 183, 399, 465, 600, 403, 364, 511, 819, 307, 847, 400, 381, 855, 961, 1240, 741, 931, 1092, 1023, 553, 1651, 781, 1815, 1240, 1281, 1093, 1767, 1953, 871, 2520, 2821, 993, 1995

Offset: 1

Walter Kehowski, Aug 10 2010

			Sigma(2^2) = 7, sigma(2^3) = 15, sigma(3^2) = 13.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (7).

Cf. A001694, A000203, A090699, A180090, A180097, A180098, A362984.

emin := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); min(L) else 0 fi end: L:=[]: for w to 1 do for n from 1 to 144 do sn:=sigma(n); if emin(n)>1 then L:=[op(L),sn]; print(n,ifactor(n),sn,ifactor(sn)) fi; od; od;

pwfQ[n_] := n == 1 || Min[Last /@ FactorInteger[n]] > 1; DivisorSigma[1, Select[ Range@ 1000, pwfQ]] (* Giovanni Resta, Feb 06 2018 *)

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, print1(sigma(k), ", "))); \\ Amiram Eldar, May 12 2023

from itertools import count, islice
from math import prod
from sympy import factorint
def A180114_gen(): # generator of terms
    for n in count(1):
        f = factorint(n)
        if all(e>1 for e in f.values()):
            yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
A180114_list = list(islice(A180114_gen(),20)) # Chai Wah Wu, May 21 2023

from math import isqrt
from sympy import mobius, integer_nthroot, divisor_sigma
def A180114(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 divisor_sigma(bisection(f, n, n)) # Chai Wah Wu, Sep 10 2024

From Amiram Eldar, May 12 2023: (Start)
Sum_{A001694(k) < x} a(k) = c * x^(3/2) + O(x^(23/18 + eps)), where c = A362984 * A090699 / 3 = 1.5572721108... (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]
Sum_{k=1..n} a(k) ~ c * n^3, where c = A362984 / (3 * A090699^2) = 0.151716514097... . (End)

a(1)=1 prepended by and more terms from Giovanni Resta, Feb 06 2018

A368710 The maximal exponent in the prime factorization of the powerful numbers.

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 2, 2, 6, 3, 4, 2, 3, 2, 3, 7, 4, 2, 2, 3, 3, 2, 5, 8, 5, 2, 4, 3, 2, 3, 4, 4, 2, 2, 3, 9, 2, 6, 4, 4, 3, 2, 6, 4, 5, 2, 5, 2, 2, 3, 5, 3, 10, 2, 3, 7, 2, 2, 4, 3, 3, 3, 2, 3, 2, 2, 5, 6, 2, 6, 2, 3, 2, 4, 5, 4, 4, 11, 2, 7, 3, 2, 8, 3, 4

Offset: 1

Amiram Eldar, Jan 04 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Generalizations of the Niven constant and the Feller-Tornier constant, International Mathematical Forum, Vol. 13, No. 9 (2018), pp. 415-425.

Cf. A001694, A033150, A048250, A051903, A059956, A090699.

Similar sequences: A368711, A368712, A368713.

s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[3000], # == 1 || Min[FactorInteger[#][[;;, 2]]] > 1 &]
(* or *)
f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1, 0, If[Min[e] > 1, Max[e], Nothing]]]; Array[f, 3000]

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1, print1(0, ", "), if(vecmin(e) > 1, print1(vecmax(e), ", "))));}

a(n) = A051903(A001694(n)).
a(n) >= 2 for n >= 2.
Sum_{a(n)<=x} = D_{2,1} * sqrt(x) + O(sqrt(x)), where D_{2,1} = (6/Pi^2) * (2 + Sum_{k>=1} (A051903(k)+2)/(sqrt(k) * A048250(k))) (Jakimczuk, 2018; Theorem 2.1 and Remark 2.3).
Asymptotic mean (consequence of the formula above): Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = D_{2,1} * zeta(3)/zeta(3/2) = D_{2,1} / A090699.
The sum in the formula for D_{2,1} converges slowly: for k up to 10^8, 10^9 and 10^10 the sums are 14.845..., 14.908... and 14.938..., respectively. Thus, a lower bound for the value of this mean, calculated by summing over k=1..10^10, is 4.738... .

A328013 Decimal expansion of the growth constant for the partial sums of powerful part of n (A057521).

Original entry on oeis.org

3, 5, 1, 9, 5, 5, 5, 0, 5, 8, 4, 1, 7, 1, 0, 6, 6, 4, 7, 1, 9, 7, 5, 2, 9, 4, 0, 3, 6, 9, 8, 5, 7, 8, 1, 7, 1, 8, 6, 0, 3, 9, 8, 0, 8, 2, 2, 5, 4, 0, 7, 8, 1, 4, 7, 1, 1, 4, 6, 4, 0, 3, 1, 4, 5, 4, 1, 7, 8, 3, 9, 8, 4, 7, 9, 7, 3, 5, 4, 0, 8, 9, 7, 7, 1, 3, 5, 8, 0, 3, 7, 5, 3, 6, 4, 6, 1, 6, 2, 0, 1, 1, 4, 5, 5

Offset: 1

Amiram Eldar, Oct 01 2019

			3.51955505841710664719752940369857817186039808225407...

D. Suryanarayana and P. Subrahmanyam, The maximal k-full divisor of an integer, Indian J. Pure Appl. Math., Vol. 12, No. 2 (1981), pp. 175-190.

Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d’un nombre, Thèse de doctorat, Université Laval (2018).
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.11, pp. 31-32.

Cf. A055231, A057521, A090699, A191622.

$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 0, -2, 0, 1}, {0, 0, 6, 0, -5}, m]; RealDigits[(1 + 2/2^(3/2) - 1/2^(5/2))*(1 + 2/3^(3/2) - 1/3^(5/2))* Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 3, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 + 2/p^3 - 1/p^5, 1/2) \\ Amiram Eldar, Jun 29 2023

The constant d1 in the paper by Cloutier et al. such that Sum_{k=1..x} A057521(k) = (d1/3)*x^(3/2) + O(x^(4/3)).
Sum_{k=1..x} 1/A055231(k) = d1*x^(1/2) + O(x^(1/3)).
Equals Product_{primes p} (1 + 2/p^(3/2) - 1/p^(5/2)).
Equals (zeta(3/2)/zeta(3)) * Product_{p prime} (1 + (sqrt(p)-1)/(p*(p-sqrt(p)+1))). - Amiram Eldar, Dec 26 2024

More terms from Vaclav Kotesovec, May 29 2020

A362975 Decimal expansion of zeta(3/4) * Product_{p prime} (1 + 1/p^(5/4) - 1/p^2 - 1/p^(9/4)) (negated).

Original entry on oeis.org

5, 8, 7, 2, 6, 1, 8, 8, 2, 0, 8, 1, 3, 8, 4, 2, 3, 9, 1, 0, 7, 4, 1, 3, 8, 1, 4, 2, 6, 6, 7, 8, 3, 5, 6, 1, 1, 4, 8, 6, 2, 6, 4, 3, 1, 1, 0, 8, 2, 9, 3, 5, 3, 5, 1, 7, 0, 7, 9, 8, 0, 4, 6, 6, 9, 0, 3, 9, 8, 2, 0, 5, 3, 5, 0, 1, 1, 2, 5, 3, 5, 6, 8, 6, 3, 3, 7, 5, 7, 9, 1, 7, 5, 1, 3, 0, 1, 2, 1, 3, 1, 6, 8, 4, 3

Offset: 1

Amiram Eldar, May 11 2023

The coefficient c_1 of the second term in the asymptotic formula for the number of cubefull numbers (A036966) not exceeding x, N(x) = c_0 * x^(1/3) + c_1 * x^(1/4) + c_2 * x^(1/5) + o(x^(1/8)) (Bateman and Grosswald, 1958; Finch, 2003).

			-5.87261882081384239107413814266783561148626431108293...

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

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.
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, A090699, A244000, A362973, A362974 (c_0), A362976 (c_2).

zeta(3/4) * prodeulerrat(1 + 1/p^5 - 1/p^8 - 1/p^9 ,1/4)

A362976 Decimal expansion of zeta(3/5) * zeta(4/5) * Product_{p prime} (1 - 1/p^(8/5) - 1/p^(9/5) - 1/p^2 + 1/p^(13/5) + 1/p^(14/5)).

Original entry on oeis.org

1, 6, 8, 2, 4, 4, 1, 5, 1, 0, 2, 3, 5, 9, 3, 2, 9, 3, 8, 9, 5, 6, 0, 0, 2, 0, 3, 4, 3, 1, 7, 7, 1, 2, 4, 5, 3, 3, 7, 2, 3, 3, 6, 2, 1, 3, 5, 7, 9, 9, 4, 9, 4, 3, 8, 5, 1, 5, 8, 3, 5, 4, 3, 9, 7, 4, 9, 6, 9, 8, 9, 7, 7, 6, 7, 6, 0, 1, 0, 6, 4, 7, 8, 5, 6, 2, 7, 7, 7, 7, 5, 4, 1, 9, 7, 6, 4, 3, 9, 5, 5, 6, 7, 5, 2

Offset: 1

Amiram Eldar, May 11 2023

The coefficient c_2 of the third term in the asymptotic formula for the number of cubefull numbers (A036966) not exceeding x, N(x) = c_0 * x^(1/3) + c_1 * x^(1/4) + c_2 * x^(1/5) + o(x^(1/8)) (Bateman and Grosswald, 1958; Finch, 2003).

			1.68244151023593293895600203431771245337233621357994...

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

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.
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, A090699, A244000, A362973, A362974 (c_0), A362975 (c_1).

zeta(3/5) * zeta(4/5) * prodeulerrat(1 - 1/p^8 - 1/p^9 - 1/p^10 + 1/p^13 + 1/p^14, 1/5)

cp's OEIS Frontend

A007913 Squarefree part of n: a(n) is the smallest positive number m such that n/m is a square.

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A362974 Decimal expansion of Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)).

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A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

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A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.

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A118896 Number of powerful numbers <= 10^n.

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A180114 a(n) = sigma(A001694(n)), sum of divisors of the powerful number A001694(n).

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