A362974 -id:A362974 - cp's OEIS frontend

A370788 Cubefull numbers with an odd number of prime factors (counted with multiplicity).

8, 27, 32, 125, 128, 243, 343, 432, 512, 648, 1331, 1728, 2000, 2048, 2187, 2197, 2592, 3125, 3888, 4913, 5000, 5488, 5832, 6859, 6912, 8000, 8192, 10125, 10368, 12167, 15552, 16807, 16875, 19208, 19683, 20000, 21296, 21952, 23328, 24389, 27000, 27648, 27783, 29791

Offset: 1

Amiram Eldar, Mar 02 2024

Jakimczuk (2024) proved:
The number of terms that do not exceed x is N(x) = c * x^(1/3) / 2 + o(x^(1/3)) where c = A362974.
The relative asymptotic density of this sequence within the cubefull numbers is 1/2.
In general, the relative asymptotic density of the s-full numbers (numbers whose exponents in their prime factorization are all >= s) with an odd number of prime factors (counted with multiplicity) within the s-full numbers is 1/2 when s is odd.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Intersection of A036966 and A026424.
Complement of A370787 within A036966.
Subsequence of A370786.
Cf. A362974.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, # > 2 &] && OddQ[Total[e]]]; Select[Range[30000], q]

is(n) = {my(e = factor(n)[, 2]); n > 1 && vecmin(e) > 2 && vecsum(e)%2;}

A362971 Partials sums of the cubefull numbers (A036966).

Original entry on oeis.org

1, 9, 25, 52, 84, 148, 229, 354, 482, 698, 941, 1197, 1540, 1972, 2484, 3109, 3757, 4486, 5350, 6350, 7374, 8670, 10001, 11729, 13673, 15673, 17721, 19908, 22105, 24506, 27098, 29842, 32967, 36342, 39798, 43686, 47686, 51782, 56695, 61695, 66879, 72367, 78199

Offset: 1

Amiram Eldar, May 13 2023

nonn

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, Theorem 2.7, p. 729.

Cf. A036966, A174172, A362974.

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

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

a(n) = Sum_{k=1..n} A036966(k).
a(n) = c * A036966(n)^(4/3) + o(A036966(n)^(4/3)), where c = A362974 / 4 = 1.1648165306... (Jakimczuk, 2017).
a(n) ~ c * n^4, where c = 1/(4 * A362974 ^ 3) = 0.002471652768... .

A371185 Indices of the cubefull numbers in the sequence of powerful numbers.

Original entry on oeis.org

1, 3, 5, 7, 8, 11, 13, 17, 18, 23, 25, 26, 30, 34, 38, 41, 42, 45, 49, 54, 55, 61, 63, 72, 77, 78, 80, 82, 83, 87, 89, 93, 99, 105, 106, 113, 115, 116, 127, 128, 130, 137, 140, 148, 151, 153, 161, 164, 166, 179, 185, 186, 188, 192, 196, 201, 206, 221, 227, 234

Offset: 1

Amiram Eldar, Mar 14 2024

			The first 5 powerful numbers are 1, 4, 8, 9, and 16. The 1st, 3rd, and 5th, 1, 8, and 16, are also cubefull numbers. Therefore, the first 3 terms of this sequence are 1, 3, and 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Cf. A001694, A036966.
Cf. A090699, A362974.
Similar sequences: A361936, A371186.

powQ[n_, emin_] := n == 1 || AllTrue[FactorInteger[n], Last[#] >= emin &]; Position[Select[Range[20000], powQ[#, 2] &], _?(powQ[#1, 3] &), Heads -> False] // Flatten

ispow(n, emin) = n == 1 || vecmin(factor(n)[, 2]) >= emin;
lista(kmax) = {my(f, c = 0); for(k = 1, kmax, if(ispow(k, 2), c++; if(ispow(k, 3), print1(c, ", "))));}

from math import isqrt, gcd
from sympy import mobius, integer_nthroot, factorint
def A371185(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 = n+x
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c -= integer_nthroot(z//y**4,3)[0]
        return c
    c, l, m = 0, 0, bisection(f,n,n)
    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 12 2024

A001694(a(n)) = A036966(n).

a(n) ~ c * n^(3/2), where c = A090699 / A362974^(3/2) = 0.216089803749...

A378485 Decimal expansion of Product_{p prime} (1 + 1/p^(5/4) + 1/p^(3/2) + 1/p^(7/4)).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 28 2024

			9.669475484382368106500666943200817938092724844476...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.6.1, p. 114.

Cf. A090699, A362974, A362975, A362976, A378486, A378487.

prodeulerrat(1 + 1/p^5 + 1/p^6 + 1/p^7, 1/4)

A378486 Decimal expansion of Product_{p prime} (1 + 1/p^(6/5) + 1/p^(7/5) + 1/p^(8/5) + 1/p^(9/5)).

Original entry on oeis.org

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

Offset: 2

Stefano Spezia, Nov 28 2024

			19.445576083900571139080089328991354647119505075485...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.6.1, p. 114.

Cf. A090699, A362974, A362975, A362976, A378485, A378487.

prodeulerrat(1 + 1/p^6 + 1/p^7 + 1/p^8 + 1/p^9, 1/5)

A378487 Decimal expansion of Product_{p prime} (1 + 1/p^(6/5) + 1/p^(7/5) - 1/p^2 - 1/p^(11/5) - 1/p^(12/5)) (negated).

Original entry on oeis.org

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

Offset: 2

Stefano Spezia, Nov 28 2024

			-16.978781483435243992799706261403133234125873342959...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.6.1, p. 114.

Cf. A090699, A362974, A362975, A362976, A378485, A378486.

zeta(4/5)*prodeulerrat(1 + 1/p^6 + 1/p^7 - 1/p^10 - 1/p^11 - 1/p^12, 1/5)

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]

cp's OEIS Frontend

A370788 Cubefull numbers with an odd number of prime factors (counted with multiplicity).

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A362971 Partials sums of the cubefull numbers (A036966).

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A371185 Indices of the cubefull numbers in the sequence of powerful numbers.

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A378485 Decimal expansion of Product_{p prime} (1 + 1/p^(5/4) + 1/p^(3/2) + 1/p^(7/4)).

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A378486 Decimal expansion of Product_{p prime} (1 + 1/p^(6/5) + 1/p^(7/5) + 1/p^(8/5) + 1/p^(9/5)).

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A378487 Decimal expansion of Product_{p prime} (1 + 1/p^(6/5) + 1/p^(7/5) - 1/p^2 - 1/p^(11/5) - 1/p^(12/5)) (negated).

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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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