A036966 -id:A036966 - cp's OEIS frontend

A046101 Biquadrateful numbers.

16, 32, 48, 64, 80, 81, 96, 112, 128, 144, 160, 162, 176, 192, 208, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 384, 400, 405, 416, 432, 448, 464, 480, 486, 496, 512, 528, 544, 560, 567, 576, 592, 608, 624, 625, 640, 648, 656, 672, 688, 704

Offset: 1

Eric W. Weisstein

nonn

The convention in the OEIS is that squareful, cubeful (A046099), biquadrateful, ... 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
Also solutions to equation tau_{-3}(n)=0, where tau_{-3} is A007428. - Enrique Pérez Herrero, Jan 19 2013
Sum_{n>0} 1/a(n)^s = Zeta(s) - Zeta(s)/Zeta(4s). - Enrique Pérez Herrero, Jan 21 2013
A051903(a(n)) > 3. - Reinhard Zumkeller, Sep 03 2015
The asymptotic density of this sequence is 1 - 1/zeta(4) = 1 - 90/Pi^4 = 0.076061... - Amiram Eldar, Jul 09 2020

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

Cf. A046100, A046099, A036967, A001694, A051903, A215267.

a046101 n = a046101_list !! (n-1)
a046101_list = filter ((> 3) . a051903) [1..]
-- Reinhard Zumkeller, Sep 03 2015

with(NumberTheory):
isBiquadrateful := n -> is(denom(Radical(n) / LargestNthPower(n, 2)) <> 1):
select(isBiquadrateful, [`$`(1..704)]);  # Peter Luschny, Jul 12 2022

lst={};Do[a=0;Do[If[FactorInteger[m][[n, 2]]>3, a=1], {n, Length[FactorInteger[m]]}];If[a==1, AppendTo[lst, m]], {m, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 15 2008 *)
Select[Range[1000],Max[Transpose[FactorInteger[#]][[2]]]>3&] (* Harvey P. Dale, May 25 2014 *)

is(n)=n>9 && vecmax(factor(n)[,2])>3 \\ Charles R Greathouse IV, Sep 03 2015

from sympy import mobius, integer_nthroot
def A046101(n):
    def f(x): return n+sum(mobius(k)*(x//k**4) for k in range(1, integer_nthroot(x,4)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

A036967 4-full numbers: if a prime p divides k then so does p^4.

Original entry on oeis.org

1, 16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 1296, 2048, 2187, 2401, 2592, 3125, 3888, 4096, 5184, 6561, 7776, 8192, 10000, 10368, 11664, 14641, 15552, 15625, 16384, 16807, 19683, 20000, 20736, 23328, 28561, 31104, 32768, 34992

Offset: 1

N. J. A. Sloane

a(m) mod prime(n) > 0 for m < A258601(n); a(A258601(n)) = A030514(n) = prime(n)^4. - Reinhard Zumkeller, Jun 06 2015

E. Kraetzel, Lattice Points, Kluwer, Chap. 7, p. 276.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 300 terms from T. D. Noe)

A030514 is a subsequence.

Cf. A001694, A036966, A046101, A258601.

import Data.Set (singleton, deleteFindMin, fromList, union)
a036967 n = a036967_list !! (n-1)
a036967_list = 1 : f (singleton z) [1, z] zs where
   f s q4s p4s'@(p4:p4s)
     | m < p4 = m : f (union (fromList $ map (* m) ps) s') q4s p4s'
     | otherwise = f (union (fromList $ map (* p4) q4s) s) (p4:q4s) p4s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a030514_list
-- Reinhard Zumkeller, Jun 03 2015

Join[{1},Select[Range[35000],Min[Transpose[FactorInteger[#]][[2]]]>3&]] (* Harvey P. Dale, Jun 05 2012 *)

is(n)=n==1 || vecmin(factor(n)[,2])>3 \\ Charles R Greathouse IV, Sep 17 2015

M(v,u,lim)=vecsort(concat(vector(#v, i, my(m=lim\v[i]); v[i]*select(t->t<=m, u))))
Gen(lim,k)={my(v=[1]); forprime(p=2, sqrtnint(lim, k), v=M(v, concat([1], vector(logint(lim,p)-k+1,e,p^(e+k-1))), lim));v}
Gen(35000,4) \\ Andrew Howroyd, Sep 10 2024

from sympy import factorint
A036967_list = [n for n in range(1,10**5) if min(factorint(n).values(),default=4) >= 4] # Chai Wah Wu, Aug 18 2021

from math import gcd
from sympy import integer_nthroot, factorint
def A036967(n):
    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 = n+x
        for u in range(1,integer_nthroot(x,7)[0]+1):
            if all(d<=1 for d in factorint(u).values()):
                for w in range(1,integer_nthroot(a:=x//u**7,6)[0]+1):
                    if gcd(w,u)==1 and all(d<=1 for d in factorint(w).values()):
                        for y in range(1,integer_nthroot(z:=a//w**6,5)[0]+1):
                            if gcd(w,y)==1 and gcd(u,y)==1 and all(d<=1 for d in factorint(y).values()):
                                c -= integer_nthroot(z//y**5,4)[0]
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) = 1.1488462139214317030108176090790939019972506733993367867997411290952527... - Amiram Eldar, Jul 09 2020

More terms from Erich Friedman

Corrected by Vladeta Jovovic, Aug 17 2002

A062838 Cubes of squarefree numbers.

Original entry on oeis.org

1, 8, 27, 125, 216, 343, 1000, 1331, 2197, 2744, 3375, 4913, 6859, 9261, 10648, 12167, 17576, 24389, 27000, 29791, 35937, 39304, 42875, 50653, 54872, 59319, 68921, 74088, 79507, 97336, 103823, 132651, 148877, 166375, 185193, 195112, 205379, 226981, 238328

Offset: 1

Jason Earls, Jul 21 2001

Cubefull numbers (A036966) all of whose nonunitary divisors are not cubefull (A362147). - Amiram Eldar, May 13 2023

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..1000

Other powers of squarefree numbers: A005117(1), A062503(2), A113849(4), A072774(all).
Cf. A000400, A036966, A225546, A362147.
A329332 column 3 in ascending order.

Select[Range[70], SquareFreeQ]^3 (* Harvey P. Dale, Jul 20 2011 *)

for(n=1,35, if(issquarefree(n),print(n*n^2)))

a(n) = my(m, c); if(n<=1, n==1, c=1; m=1; while(cAltug Alkan, Dec 03 2015

from math import isqrt
from sympy import mobius
def A062838(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**3 # Chai Wah Wu, Sep 11 2024

A055229(a(n)) = A005117(n) and A055229(m) < A005117(n) for m < a(n). - Reinhard Zumkeller, Apr 09 2010
a(n) = A005117(n)^3. - R. J. Mathar, Dec 03 2015
{a(n)} = {A225546(A000400(n)) : n >= 0}, where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Oct 31 2019
Sum_{n>=1} 1/a(n) = zeta(3)/zeta(6) = 945*zeta(3)/Pi^6 (A157289). - Amiram Eldar, May 22 2020

More terms from Dean Hickerson, Jul 24 2001

More terms from Vladimir Joseph Stephan Orlovsky, Aug 15 2008

A262675 Exponentially evil numbers.

Original entry on oeis.org

1, 8, 27, 32, 64, 125, 216, 243, 343, 512, 729, 864, 1000, 1024, 1331, 1728, 1944, 2197, 2744, 3125, 3375, 4000, 4096, 4913, 5832, 6859, 7776, 8000, 9261, 10648, 10976, 12167, 13824, 15552, 15625, 16807, 17576, 19683, 21952, 23328, 24389, 25000, 27000, 27648, 29791

Offset: 1

Vladimir Shevelev, Sep 27 2015

Or the numbers whose prime power factorization contains primes only in evil exponents (A001969): 0, 3, 5, 6, 9, 10, 12, ...
If n is in the sequence, then n^2 is also in the sequence.
A268385 maps each term of this sequence to a unique nonzero square (A000290), and vice versa. - Antti Karttunen, May 26 2016

			864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Subsequence of A036966.
Apart from 1, a subsequence of A270421.
Indices of ones in A270418.
Sequence A270437 sorted into ascending order.
Cf. A001969, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A010059, A124010, A268385, A270428.

a262675 n = a262675_list !! (n-1)
a262675_list = filter
   (all (== 1) . map (a010059 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 25 2015

{1}~Join~Select[Range@ 30000, AllTrue[Last /@ FactorInteger[#], EvenQ@ First@ DigitCount[#, 2] &] &] (* Michael De Vlieger, Sep 27 2015, Version 10 *)
expEvilQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], EvenQ[DigitCount[#, 2, 1]] &]; With[{max = 30000}, Select[Union[Flatten[Table[i^2*j^3, {j, Surd[max, 3]}, {i, Sqrt[max/j^3]}]]], expEvilQ]] (* Amiram Eldar, Dec 01 2023 *)

isok(n) = {my(f = factor(n)); for (i=1, #f~, if (hammingweight(f[i,2]) % 2, return (0));); return (1);} \\ Michel Marcus, Sep 27 2015

use ntheory ":all"; sub isok { my @f = factor_exp($[0]); return scalar(grep { !(hammingweight($->[1]) % 2) } @f) == @f; } # Dana Jacobsen, Oct 26 2015

Product_{k=1..A001221(n)} A010059(A124010(n,k)) = 1. - Reinhard Zumkeller, Oct 25 2015

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^A001969(k)) = Product_{p prime} f(1/p) = 1.2413599378..., where f(x) = (1/(1-x) + Product_{k>=0} (1 - x^(2^k)))/2. - Amiram Eldar, May 18 2023, Dec 01 2023

More terms from Michel Marcus, Sep 27 2015

A335988 Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.

Original entry on oeis.org

1, 8, 27, 32, 125, 128, 216, 243, 343, 512, 864, 1000, 1331, 1944, 2048, 2187, 2197, 2744, 3125, 3375, 3456, 4000, 4913, 6859, 7776, 8192, 9261, 10648, 10976, 12167, 13824, 16000, 16807, 17496, 17576, 19683, 24389, 25000, 27000, 29791, 30375, 31104, 32768, 35937

Offset: 1

Amiram Eldar, Jul 03 2020

nonn

This sequence is a permutation of A355038.
This sequence is also a permutation of the exponentially odd numbers (A268335) multiplied by the square of their squarefree kernel (A007947).
a(n)/rad(a(n)) is a permutation of the squares.
a(n)/rad(a(n))^2 is a permutation of the exponentially odd numbers.

			8 = 2^3 is a term since the exponent of its prime factor 2 is 3 which is odd and larger than 1.

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

Intersection of A001694 and A268335.
Intersection of A036966 and A268335.
A355038 in ascending order.
A030078, A050997, A092759, A179665, A079395 and A138031 are subsequences.
Cf. A007947, A065487.

Join[{1}, Select[Range[10^5], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]

from math import isqrt, prod
from sympy import factorint
def afind(N): # all terms up to limit N
    cands = (n**2*prod(factorint(n**2)) for n in range(1, isqrt(N//2)+2))
    return sorted(c for c in cands if c <= N)
print(afind(4*10**4)) # Michael S. Branicky, Jun 16 2022

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911... (A065487).

A337050 Numbers without an exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Amiram Eldar, Aug 12 2020

Numbers k such that the powerful part (A057521) of k is a cubefull number (A036966).
Numbers k such that A003557(k) = k/A007947(k) is a powerful number (A001694).
The asymptotic density of this sequence is Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).
A304364 is apparently a subsequence.
These numbers were named semi-2-free integers by Suryanarayana (1971). - Amiram Eldar, Dec 29 2020

			6 = 2^1 * 3^1 is a term since none of the exponents in its prime factorization is equal to 2.
9 = 3^2 is not a term since it has an exponent 2 in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.

Complement of A038109.
Cf. A003557, A007947, A057521, A304364, A330596.
A005117, A036537, A036966, A048109, A175496, A268335 and A336590 are subsequences.
Numbers without an exponent k in their prime factorization: A001694 (k=1), this sequence (k=2), A386799 (k=3), A386803 (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: this sequence (m=0), A386796 (m=1), A386797 (m=2), A386798 (m=3).

q:= n-> andmap(i-> i[2]<>2, ifactors(n)[2]):
select(q, [$1..100])[];  # Alois P. Heinz, Aug 12 2020

Select[Range[100], !MemberQ[FactorInteger[#][[;;, 2]], 2] &]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 2, return(0))); 1; } \\ Amiram Eldar, Oct 21 2023

Sum_{n>=1} 1/a(n)^s = zeta(s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)), for s > 1. - Amiram Eldar, Oct 21 2023

A065487 Decimal expansion of Product_{p prime} (1 + 1/(p*(p^2-1))).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

			1.2312911488886035627747876512720337...

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A000010, A001615, A003557, A036966, A047994, A072802.

$MaxExtraPrecision = 600; digits = 99; terms = 600; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0}, LinearRecurrence[{0, 2, -1, -1, 1}, {3, 0, 5, -3, 7}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + 1/(p*(p^2-1))) \\ Amiram Eldar, Mar 17 2021

Equals Sum_{k>=1} 1/A001615(A036966(k)). - Amiram Eldar, Jun 23 2020
Equals Sum_{k>=1} A003557(k)/k^3. - Amiram Eldar, Jan 25 2024
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A047994(k)/A000010(k). - Amiram Eldar, Feb 04 2024

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).

A065483 Decimal expansion of totient constant Product_{p prime} (1 + 1/(p^2*(p-1))).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

The sum of the reciprocals of the cubefull numbers (A036966). - Amiram Eldar, Jun 23 2020

			1.339784153574347246599152586514886052775...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Eric Weisstein's World of Mathematics, Totient Summatory Function

Cf. A036966, A065484, A078074.

$MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0}, LinearRecurrence[{2, -1, -1, 1}, {3, 4, 5, 3}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 + 1/(p^2*(p-1))) \\ Vaclav Kotesovec, Sep 19 2020

Equals (6/Pi^2) * A065484. - Amiram Eldar, Jun 23 2020

A384051 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cubefull number.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 8, 8, 4, 10, 6, 12, 6, 8, 16, 16, 8, 18, 12, 12, 10, 22, 16, 24, 12, 27, 18, 28, 8, 30, 32, 20, 16, 24, 24, 36, 18, 24, 32, 40, 12, 42, 30, 32, 22, 46, 32, 48, 24, 32, 36, 52, 27, 40, 48, 36, 28, 58, 24, 60, 30, 48, 64, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A384040.
Cf. A036966, A360539, A360540, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), this sequence (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[e < 3, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] < 3, 1, 0));}

Multiplicative with a(p^e) = p^e-1 if e <= 2, and p^e if e >= 3.
a(n) = n * A047994(n) / A384049(n).
a(n) = A047994(A360539(n)) * A360540(n).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5) = 0.714093594477970831206... .

cp's OEIS Frontend

A046101 Biquadrateful numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

A036967 4-full numbers: if a prime p divides k then so does p^4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Python

Formula

Extensions

A062838 Cubes of squarefree numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A262675 Exponentially evil numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Perl

Formula

Extensions

A335988 Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A337050 Numbers without an exponent 2 in their prime factorization.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs