A062838 -id:A062838 - cp's OEIS frontend

A374459 Nonsquarefree exponentially odd numbers.

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536

Offset: 1

Amiram Eldar, Jul 09 2024

First differs from A301517 at n = 1213. A301517(1213) = 12500 = 2^2 * 5^5 is not an exponentially odd number.
Numbers whose exponents in their prime factorization are all odd and at least one of them is larger than 1.
All the squarefree numbers (A005117) are exponentially odd. Therefore, the sequence of exponentially odd numbers (A268335) is a disjoint union of the squarefree numbers and this sequence.
The asymptotic density of this sequence is A065463 - A059956 = 0.096515099145... .

			8 = 2^3 is a term since 3 is odd and larger than 1.

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

Intersection of A013929 (or A046099) and A268335.
Subsequence of A301517.
Subsequences: A062838 \ {1}, A065036, A102838, A113850, A113852, A179671, A190011, A335988 \ {1}.
Cf. A005117, A374460.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[1000], q]

is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}

a(n) = A268335(A374460(n)).

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

A368248 The number of unitary divisors of the cubefull part of n (A360540).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 19 2023

First differs from A061704 and A362852 at n = 64, and from A304327 at n = 72.
Also, the number of squarefree divisors of the cubefull part of n.
Also, the number of cubes of squarefree numbers (A062838) that divide n.
The number of unitary divisors of n that are cubefull numbers (A036966). - Amiram Eldar, Jun 19 2025

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

Cf. A004709, A034444, A036966, A062838, A157289, A307428, A323308, A360540, A366145.

Cf. A061704, A304327, A362852.

f[p_, e_] := If[e > 2, 2, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x < 3, 1, 2), factor(n)[, 2]));

a(n) = A034444(A360540(n)).
a(n) = abs(A307428(n)).
Multiplicative with a(p) = 1 for e <= 2, and 2 for e >= 3.
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A034444(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(6*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(6) = 1.181564... (A157289).
In general, the asymptotic mean of the number of unitary divisors of the k-full part of n is zeta(k)/zeta(2*k).

A336591 Numbers whose exponents in their prime factorization are either 1, 3, or both.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

The asymptotic density of this sequence is zeta(6)/(zeta(2) * zeta(3)) * Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5) = 0.68428692418686231814196872579121808347231273672316377728461822629005... (Cohen, 1962).

First differs from A036537 at n = 89. A036537(89) = 128 = 2^7 is not a term of this sequence.

			1 is a term since it has no exponents, and thus it has no exponent that is not 1 or 3.
2 is a term since 2 = 2^1 has only the exponent 1 in its prime factorization.
24 is a term since 24 = 2^3 * 3^1 has the exponents 1 and 3 in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, No. 12, Vol. 1 (1962), pp. 77-84.

Intersection of A046100 and A036537.
Intersection of A046100 and A268335.
A005117 and A062838 are subsequences.
Cf. A068468.

seqQ[n_] := AllTrue[FactorInteger[n][[;;,2]], MemberQ[{1, 3}, #] &]; Select[Range[100], seqQ]

from itertools import count, islice
from sympy import factorint
def A336591_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e==1 or e==3 for e in factorint(n).values()),count(max(startvalue,1)))
A336591_list = list(islice(A336591_gen(),20)) # Chai Wah Wu, Jun 22 2023

A153147 a(n) = A007916(n)^3.

Original entry on oeis.org

8, 27, 125, 216, 343, 1000, 1331, 1728, 2197, 2744, 3375, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 17576, 21952, 24389, 27000, 29791, 35937, 39304, 42875, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336, 103823

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

2^3=8, 3^3=27, 4^3=64=2^6 is not in the sequence, 5^3=125, 6^3=216, ...

Cf. A000578, A007916, A153157, A153158, A153159, A153160, A062838.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^3

from sympy import mobius, integer_nthroot
def A153147(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**3

Edited by Ray Chandler, Dec 22 2008

A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2.

Original entry on oeis.org

32, 72, 108, 128, 200, 243, 256, 288, 392, 432, 500, 512, 576, 648, 675, 800, 864, 968, 972, 1024, 1125, 1152, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1944, 2000, 2048, 2187, 2304, 2312, 2592, 2700, 2888, 2916, 3087, 3125, 3136, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4096, 4232, 4500, 4563, 4608

Offset: 1

V. Raman, Sep 07 2012

nonn

Powerful numbers (A001694) that are not squares of cubefree numbers (A004709), cubes of squarefree numbers (A062838), or 6th powers of primes (A030516). - Amiram Eldar, Feb 07 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A143610, A216426.

Cf. A001694, A002117, A004709, A013661, A013664, A062838, A085966.

With[{max = 5000}, Union[Table[i^2*j^3, {j, 2, max^(1/3)}, {i, 2, Sqrt[max/j^3]}] // Flatten]] (* Amiram Eldar, Feb 07 2023 *)

list(lim)=my(v=List()); for(b=2, sqrtnint(lim\4, 3), for(a=2, sqrtint(lim\b^3), listput(v, a^2*b^3))); Set(v) \\ Charles R Greathouse IV, Jan 03 2014

from math import isqrt
from sympy import mobius, integer_nthroot, primepi
def A216427(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):
        j, b = isqrt(x), integer_nthroot(x,6)[0]
        l, c = 0, n+x-1+primepi(b)+sum(mobius(k)*(j//k**3) for k in range(1, b+1))
        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)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 13 2024

Sum_{n>=1} 1/a(n) = 1 + ((zeta(2)-1)*(zeta(3)-1)-1)/zeta(6) - P(6) = 0.12806919584708298724..., where P(s) is the prime zeta function. - Amiram Eldar, Feb 07 2023

A175496 Positive integers k where k is not squarefree but the number of divisors of k is a power of 2.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 88, 104, 120, 125, 128, 135, 136, 152, 168, 184, 189, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 375, 376, 378, 384, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 640

Offset: 1

Leroy Quet, May 30 2010

nonn

This sequence is not the same as sequence A048109.

The asymptotic density of this sequence is A327839 - A059956 = 0.0799000375... - Amiram Eldar, Nov 01 2020

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

Cf. A059956, A327839.
Complement of A005117 within A036537.
Subsequence of A013929.
A048109, A062838\{1}, A113852 are subsequences.

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; aQ[n_] := ! SquareFreeQ[n] && pow2Q[DivisorSigma[0, n]]; Select[Range[640], aQ] (* Amiram Eldar, Sep 21 2019 *)

isok(k) = my(d=numdiv(k)); !issquarefree(k) && (2^valuation(d, 2) == d); \\ Michel Marcus, Nov 20 2020

from itertools import count, islice
from sympy import factorint
def A175496_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:max(f:=factorint(n).values())>1 and all(map(lambda m:not((k:=m+1)&-k)^k,f)),count(max(startvalue,2)))
A175496_list = list(islice(A175496_gen(),30)) # Chai Wah Wu, Jan 04 2023

Extended by D. S. McNeil, May 31 2010

A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

72, 108, 200, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 1800, 2312, 2700, 2888, 3087, 3267, 3528, 4232, 4500, 4563, 5292, 5324, 5400, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10584, 10952, 11979, 12168, 12348, 13068, 13448, 13500, 14283, 14792

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {2, 3}.

Number k such that A051904(k) = 2 and A051903(k) = 3.

Equals A338325 \ (A062503 UNION A062838).
Subsequence of A001694 and A046100.
A143610 is a subsequence.
Cf. A051903, A051904, A136568.
Cf. A082020, A157289, A330595.

Select[Range[15000], Union[FactorInteger[#][[;; , 2]]] == {2, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [2, 3];
```

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) - 15/Pi^2 - zeta(3)/zeta(6) + 1 = A330595 - A082020 - A157289 + 1 = 0.047550294197921818806... .

A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

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

Cf. A001222, A001694 (positions of nonzero terms), A005361, A007427, A008683, A008836, A028242, A052485 (positions of 0's), A062838 (positions of 1's), A112526, A252505, A322483, A332685, A332713.

Table[Sum[MoebiusMu[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 100}]
A005361[n_] := Times @@ (#[[2]] & /@ FactorInteger[n]); a[n_] := Sum[(-1)^PrimeOmega[n/d] A005361[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
f[p_, e_] := 3*Floor[e/2] - e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

a(n) = sumdiv(n, d, moebius(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(2*s)^2 * zeta(3*s) / zeta(6*s).
a(n) = Sum_{d|n} mu(lcm(d, n/d)/d).
a(n) = Sum_{d|n} (-1)^bigomega(n/d) * A005361(d).
a(n) = Sum_{d|n} A010052(n/d) * A112526(d).
Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)*log(n)/(2*zeta(3)) + ((2*gamma - 1)*zeta(3/2) + 3*zeta'(3/2)/2 - 3*zeta(3/2)*zeta'(3)/zeta(3)) * sqrt(n)/zeta(3) + 6*zeta(2/3)^2 * n^(1/3)/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 21 2020
Multiplicative with a(p^e) = A028242(e). - Amiram Eldar, Nov 30 2020

A370240 The sum of divisors of n that are cubes of squarefree numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 28, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 13 2024

First differs from A366904 at n = 32, and from A113061 at n = 64.

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

Cf. A062838, A113061, A366904, A370239.

f[p_, e_] := If[e <= 2, 1, 1 + p^3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = 1 for e <= 2, and a(p^e) = 1 + p^3 for e >= 3.
Dirichlet g.f.: zeta(s)*zeta(3*s-3)/zeta(6*s-6).
Sum_{k=1..n} a(k) ~ c * n^(4/3) + n, where c = 3*zeta(4/3)/(2*Pi^2) = 0.5472769126... .

A375076 Numbers whose prime factorization exponents include at least one 1, at least one 3 and no other exponents.

Original entry on oeis.org

24, 40, 54, 56, 88, 104, 120, 135, 136, 152, 168, 184, 189, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 344, 351, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 664, 680, 686, 696, 702, 712, 728, 744, 750

Offset: 1

Amiram Eldar, Jul 29 2024

First differs from its subsequence A360793 at n = 79: a(79) = 1080 = 2^3 * 3^3 * 5 is not a term of A360793.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 3}.
The asymptotic density of this sequence is ((zeta(6)/zeta(3)) * Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5) - 1)/zeta(2) = 0.076359822332835689478... .

Equals A336591 \ (A005117 UNION A062838).
Subsequences: A065036, A360793.
Cf. A002117, A013661, A013664, A136568.

Select[Range[750], Union[FactorInteger[#][[;; , 2]]] == {1, 3} &]

PARI
```
is(k) = Set(factor(k)[,2]) == [1, 3];
```

cp's OEIS Frontend

A374459 Nonsquarefree exponentially odd numbers.

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A368248 The number of unitary divisors of the cubefull part of n (A360540).

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A336591 Numbers whose exponents in their prime factorization are either 1, 3, or both.

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A153147 a(n) = A007916(n)^3.

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A216427 Numbers of the form a^2*b^3, where a >= 2 and b >= 2.

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A175496 Positive integers k where k is not squarefree but the number of divisors of k is a power of 2.

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A375073 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no other exponents.

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