A036966 -id:A036966 - cp's OEIS frontend

A366145 The number of divisors of the largest divisor of n that is a cubefull number (A036966).

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 01 2023

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

Cf. A000005, A036966, A357669, A360540, A366076, A366146, A366147.

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

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

a(n) = A000005(A360540(n)).
a(n) = A000005(n)/A366147(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A000005(n), with equality if and only if n is cubefull (A036966).
Multiplicative with a(p^e) = 1 if e <= 2 and e+1 otherwise.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 3/p^(3*s) - 2/p^(4*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 + 1/p^4 - 2/p^5) = 1.76434793373691907811... .

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

A366076 The number of prime factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Sep 28 2023

The sum of exponents larger than 2 in the prime factorization of n.

The number of distinct prime factors of the largest divisor of n that is a cubefull number is A295659(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Sums of omega(n) and Omega(n) over the k-free parts and k-full parts of some particular sequences, Integers, Vol. 22 (2022), Article #A113.

Cf. A001222, A004709, A036966, A046100, A085541, A152441, A295659, A360540, A366077.

Similar sequence: A275812 (number of prime factors of the powerful part).

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

a(n) = vecsum(apply(x -> if(x < 3, 0, x), factor(n)[, 2]));

a(n) = A001222(A360540(n)).
a(n) = A001222(n) - A366077(n).
Additive with a(p^e) = 0 if e <= 2, and a(p^e) = e for e >= 3.
a(n) >= 0, with equality if and only if n is cubefree (A004709).
a(n) <= A001222(n), with equality if and only if n is cubefull (A036966).
a(n) >= 3*A295659(n), with equality if and only if n is a biquadratefree number (A046100).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2/p^3 + 1/(p^2*(p-1))) = 2 * A085541 + A152441 = 0.67043452760761670220... .

A371413 Dedekind psi function applied to the cubefull numbers (A036966).

Original entry on oeis.org

1, 12, 24, 36, 48, 96, 108, 150, 192, 432, 324, 384, 392, 864, 768, 750, 1296, 972, 1728, 1800, 1536, 2592, 1452, 3456, 3888, 3600, 3072, 2916, 2366, 2744, 5184, 4704, 3750, 5400, 6912, 7776, 7200, 6144, 5202, 9000, 10368, 9408, 11664, 8748, 7220, 13824, 15552

Offset: 1

Amiram Eldar, Mar 22 2024

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

Cf. A001615, A036966, A065464.

Similar sequences: A323332, A371412, A371415.

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Join[{1}, psi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]
(* or *)
f[n_] := Module[{f = FactorInteger[n], p, e}, If[n == 1, 1, p = f[[;;, 1]]; e = f[[;;, 2]]; If[Min[e] > 2, Times @@ ((p+1) * p^(e-1)), Nothing]]]; Array[f, 20000]

dedpsi(f) = prod(i = 1, #f~, (f[i, 1] + 1) * f[i, 1]^(f[i, 2]-1));
lista(max) = {my(f); print1(1, ", "); for(k = 2, max, f = factor(k); if(vecmin(f[, 2]) > 2, print1(dedpsi(f), ", "))); }

a(n) = A001615(A036966(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p^2-1)*p)) = 1.231291... (A065487).

A336593 Numbers k such that k/A008835(k) is cubeful (A036966), where A008835(k) is the largest 4th power dividing k.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 128, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 384, 392, 408, 424, 432, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

Numbers such that at least one of the exponents in their prime factorization is of the form 4*m + 3.
The asymptotic density of this sequence is 1 - zeta(4)/zeta(3) = 0.0996073223... (Cohen, 1963).
The number of divisors of all the terms is divisible by 4.

			8 is a term since 8 = 2^3 and 3 is of the form 4*m + 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.

Complement of A336592.
Complement of A336594 within A252849.
Cf. A002117, A008835, A013662, A036966.
A176297 is a subsequence.

Select[Range[540], Max[Mod[FactorInteger[#][[;; , 2]], 4]] == 3 &]

A366146 The sum of divisors of the largest divisor of n that is a cubefull number (A036966).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 40, 1, 1, 1, 1, 63, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 40, 1, 15, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 31, 121, 1

Offset: 1

Amiram Eldar, Oct 01 2023

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

Cf. A000203, A036966, A295294, A360540, A366076, A366145, A366148.

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

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

a(n) = A000203(A360540(n)).
a(n) = A000203(n)/A366148(n).
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A000203(n), with equality if and only if n is cubefull (A036966).
Multiplicative with a(p^e) = 1 if e <= 2 and (p^(e+1)-1)/(p-1) otherwise.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(3*s-3) + 1/p^(3*s-2) + 1/p^(3*s-1) - 1/p^(4*s-3) - 1/p^(4*s-2)).

A371412 Euler totient function applied to the cubefull numbers (A036966).

Original entry on oeis.org

1, 4, 8, 18, 16, 32, 54, 100, 64, 72, 162, 128, 294, 144, 256, 500, 216, 486, 288, 400, 512, 432, 1210, 576, 648, 800, 1024, 1458, 2028, 2058, 864, 1176, 2500, 1800, 1152, 1296, 1600, 2048, 4624, 2000, 1728, 2352, 1944, 4374, 6498, 2304, 2592, 3200, 4096, 5292, 4000

Offset: 1

Amiram Eldar, Mar 22 2024

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

Cf. A000010, A013661, A036966, A098198.

Similar sequences: A323333, A358039, A371413, A371414.

Join[{1}, EulerPhi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]
(* or *)
f[n_] := Module[{f = FactorInteger[n], p, e}, If[n == 1, 1, p = f[[;;, 1]]; e = f[[;;, 2]]; If[Min[e] > 2, Times @@ ((p-1) * p^(e-1)), Nothing]]]; Array[f, 20000]

lista(max) = {my(f); print1(1, ", "); for(k = 2, max, f = factor(k); if(vecmin(f[, 2]) > 2, print1(eulerphi(f), ", ")));}

a(n) = A000010(A036966(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p-1)^2*p)) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^3 + 3/p^4 + 1/p^5) = 1.65532418864085918623... .

A360541 a(n) is the least number k such that k*n is a cubefull number (A036966).

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225, 1, 289, 12, 361, 50, 441, 484, 529, 9, 5, 676, 1, 98, 841, 900, 961, 1, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 9, 7, 20, 2601, 338, 2809, 4, 3025, 49, 3249, 3364

Offset: 1

Amiram Eldar, Feb 11 2023

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

Cf. A036966, A329376, A356193, A360539.

f[p_, e_] := p^(Max[e, 3] - e); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

a(n) = 1 if and only if n is cubefull number (A036966).
a(n) = A356193(n)/n.
a(n) = A360539(n)^2/A329376(n)^3.
Multiplicative with a(p^e) = p^(max(e, 3) - e), i.e., a(p) = p^2, a(p^2) = p, and a(p^e) = 1 for e >= 3.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(2-s) - p^(-s) - p^(2-2*s) + p^(1-2*s) - p^(1-3*s) + p^(-3*s)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.2078815423... .

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... .

A362985 Decimal expansion of the asymptotic mean of the abundancy index of the cubefull numbers (A036966).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 12 2023

			2.48217919642235952546167643674687698536368940971930468354...

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. (6).

Cf. A000203, A036966, A017665, A017666, A362986.

Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A245058 (even), A240976 (squares), A306633 (squarefree), A362984 (powerful).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -1, -2, 3, -2, -1, 3, -2, -2, 3, -1, -2, 3, -1, -1, 1}, {0, 0, 0, -4, 0, 6, 7, 4, 9, 0, -11, -22, -26, -21, -15, 20}, m]; RealDigits[((2^5 + 2^(10/3) + 2^3 + 2^(8/3) - 1)/(2^(10/3)*(2^(5/3) + 2^(1/3) + 1)))*((3^5 + 3^(10/3) + 3^3 + 3^(8/3) - 1)/(3^(10/3)*(3^(5/3) + 3^(1/3) + 1))) * Zeta[4/3] * 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]]

zeta(4/3) * prodeulerrat((p^15 + p^10 + p^9 + p^8 - 1)/(p^10 * (p^5 + p + 1)), 1/3)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A362986(k)/A036966(k).

Equals zeta(4/3) * Product_{p prime} ((p^5 + p^(10/3) + p^3 + p^(8/3) - 1)/(p^(10/3) * (p^(5/3) + p^(1/3) + 1))).

cp's OEIS Frontend

A366145 The number of divisors of the largest divisor of n that is a cubefull number (A036966).

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

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A366076 The number of prime factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.

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A371413 Dedekind psi function applied to the cubefull numbers (A036966).

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A336593 Numbers k such that k/A008835(k) is cubeful (A036966), where A008835(k) is the largest 4th power dividing k.

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A366146 The sum of divisors of the largest divisor of n that is a cubefull number (A036966).

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A371412 Euler totient function applied to the cubefull numbers (A036966).

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A360541 a(n) is the least number k such that k*n is a cubefull number (A036966).

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