A362974 -id:A362974 - cp's OEIS frontend

A051904 Minimal exponent in prime factorization of n.

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Dec 16 1999

The asymptotic mean of this sequence is 1 (Niven, 1969). - Amiram Eldar, Jul 10 2020

Let k = A007947(n), then for n > 1 k^a(n) is the greatest power of k which divides n; see example. - David James Sycamore, Sep 07 2023

			For n = 72 = 2^3*3^2, a(72) = min(exponents) = min(3,2) = 2.
For n = 72, using alternative definition: rad(72) = 6; and 6^2 = 36 divides 72 but no higher power of 6 divides 72, so a(72) = 2.
For n = 432, rad(432) = 6 and 6^3 = 216 divides 432 but no higher power of 6 divides 432, therefore a(432) = 3. - _David James Sycamore_, Sep 08 2023

Daniel Forgues, Table of n, a(n) for n = 1..100000
Cao Hui-Zhong, The Asymptotic Formulas Related to Exponents in Factoring Integers, Math. Balkanica, Vol. 5 (1991), Fasc. 2.
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant
Index entries for sequences computed from exponents in factorization of n

Cf. A000961, A001221, A005361, A008479, A051903, A052409, A076558, A124010, A362974.

Cf. A007947.

a051904 1 = 0
a051904 n = minimum $ a124010_row n  -- Reinhard Zumkeller, Jul 15 2012

a := proc (n) if n = 1 then 0 else min(seq(op(2, op(j, op(2, ifactors(n)))), j = 1 .. nops(op(2, ifactors(n))))) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, May 20 2015

Table[If[n == 1, 0, Min @@ Last /@ FactorInteger[n]], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=vecmin(factor(n)[,2]) \\ Charles R Greathouse IV, Nov 19 2012

from sympy import factorint
def a(n):
    f = factorint(n)
    l = [f[p] for p in f]
    return 0 if n == 1 else min(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 13 2017

(define (A051904 n) (cond ((= 1 n) 0) ((= 1 (A001221 n)) (A001222 n)) (else (min (A067029 n) (A051904 (A028234 n)))))) ;; Antti Karttunen, Jul 12 2017

a(n) = min_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(1) = 0, for n > 1, if A001221(n) = 1 (when n is in A000961), a(n) = A001222(n), otherwise a(n) = min(A067029(n), a(A028234(n))). - Antti Karttunen, Jul 12 2017
Sum_{k=1..n} a(k) ~ n + zeta(3/2)*n^(1/2)/zeta(3) + (zeta(2/3)/zeta(2) + c0)*n^(1/3), where c0 = A362974 = Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)) [Cao Hui-Zhong, 1991]. - Vaclav Kotesovec, Mar 24 2025

A337736 The number of cubefull numbers (A036966) between the consecutive cubes n^3 and (n+1)^3.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 17 2020

nonn

For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1991).

			a(2) = 1 since there is one cubefull number, 16 = 2^4, between 2^3 = 8 and 3^3 = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, The distribution of cube-full numbers, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295. See section 3, p. 291.

Cf. A000578, A036966, A119241, A337737, A362974.

cubQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 2; a[n_] := Count[Range[n^3 + 1, (n + 1)^3 - 1], _?cubQ]; Array[a, 100]

from math import gcd
from sympy import integer_nthroot, factorint
def A337736(n):
    def f(x):
        c = 0
        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
    return f((n+1)**3-1)-f(n**3) # Chai Wah Wu, Sep 10 2024

Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = A362974 - 1 = 3.659266... . - Amiram Eldar, May 11 2023

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)

A362986 a(n) = A000203(A036966(n)), the sum of divisors of the n-th cubefull number A036966(n).

Original entry on oeis.org

1, 15, 31, 40, 63, 127, 121, 156, 255, 600, 364, 511, 400, 1240, 1023, 781, 1815, 1093, 2520, 2340, 2047, 3751, 1464, 5080, 5460, 4836, 4095, 3280, 2380, 2801, 7623, 6000, 3906, 6240, 10200, 11284, 9828, 8191, 5220, 11715, 15367, 12400, 16395, 9841, 7240, 20440

Offset: 1

Amiram Eldar, May 12 2023

nonn

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. A000203, A036966, A180114, A362974, A362985.

DivisorSigma[1, Select[Range[10^4], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(sigma(k), ", ")));

from itertools import count, islice
from math import prod
from sympy import factorint
def A362986_gen(): # generator of terms
    for n in count(1):
        f = factorint(n)
        if all(e>2 for e in f.values()):
            yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
A362986_list = list(islice(A362986_gen(),20)) # Chai Wah Wu, May 21 2023

Sum_{A036966(k) < x} a(k) = c * x^(4/3) + O(x^(113/96 + eps)), where c = A362985 * A362974 / 4 = 2.8912833599... (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

Sum_{k=1..n} a(k) ~ c * n^4, where c = A362985 / (4 * A362974^3) = 0.006135085083... .

A376363 The number of distinct prime factors of the cubefull numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, Distribution of omega(n) over h-free and h-full numbers, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.2.
Index entries for sequences related to powerful numbers.

Cf. A001221, A036966, A072047, A077761, A362974, A376361, A376364, A376365.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 2 &], Length[e], Nothing]]; Array[f, 60000]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] < 3, is = 0; break)); if(is, print1(#e, ", ")));}

a(n) = A001221(A036966(n)).

Sum_{A036966(k) <= x} a(k) = c * x^(1/3) * (log(log(x)) + B - log(3) + L(3, 4) - L(3, 6)) + O(x^(1/3)/log(x)), where c = A362974, B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), L(3, 4) = 1.65235055631578303808..., and L(3, 6) = 0.67060646664392140547... (Das et al., 2024).

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

A371186 Indices of the cubes in the sequence of cubefull numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 13, 15, 18, 20, 23, 24, 29, 32, 34, 38, 39, 43, 45, 48, 50, 54, 57, 58, 61, 67, 69, 73, 75, 77, 81, 85, 88, 90, 94, 96, 99, 102, 105, 107, 110, 113, 117, 124, 126, 128, 130, 135, 137, 139, 143, 147, 149, 153, 158, 160, 163, 167, 169, 172, 176

Offset: 1

Amiram Eldar, Mar 14 2024

Equivalently, the number of cubefull numbers that do not exceed n^3.
The asymptotic density of this sequence is 1 / A362974 = 0.214626074... .
If k is a term of A371187 then a(k) and a(k+1) are consecutive integers, i.e., a(k+1) = a(k) + 1.

			The first 4 cubefull numbers are 1, 8, 16, and 27. The 1st, 2nd, and 4th, 1, 8, and 27, are the first 3 cubes. Therefore, the first 3 terms of this sequence are 1, 2, and 4.

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

Cf. A000578, A036966, A337736, A362974, A371187.

Similar sequences: A361936, A371185.

cubQ[n_] := n == 1 || AllTrue[FactorInteger[n], Last[#] >= 3 &]; Position[Select[Range[10^6], cubQ], _?(IntegerQ[Surd[#1, 3]] &)] // Flatten
(* or *)
seq[max_] := Module[{cubs = Union[Flatten[Table[i^5*j^4*k^3, {i, 1, Surd[max, 5]}, {j, 1, Surd[max/i^5, 4]}, {k, Surd[max/(i^5*j^4), 3]}]]], s = {}}, Do[If[IntegerQ[Surd[cubs[[k]], 3]], AppendTo[s, k]], {k, 1, Length[cubs]}]; s]; seq[10^6]

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

A036966(a(n)) = A000578(n) = n^3.
a(n+1) - a(n) = A337736(n) + 1.
a(n) ~ c * n, where c = A362974.

A376364 The number of unitary divisors that are cubes of primes applied to the cubefull numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, On the number of prime factors with a given multiplicity over h-free and h-full numbers, Journal of Number Theory, Vol. 267 (2025), pp. 176-201; arXiv preprint, arXiv:2409.11275 [math.NT], 2024. See Theorem 1.3.
Index entries for sequences related to powerful numbers.

Cf. A036966, A077761, A295883, A362974, A376362, A376363.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 2 &], Count[e, 3], Nothing]]; Array[f, 60000]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] < 3, is = 0; break)); if(is, print1(#select(x -> x == 3, e), ", ")));}

a(n) = A295883(A036966(n)).

Sum_{A036966(k) <= x} a(k) = c * sqrt(x) * (log(log(x)) + B - log(3) - L(3, 6)) + O(x^(1/3)/log(x)), where c = A362974, B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), and L(3, 6) = 0.67060646664392140547... (Das et al., 2025).

A370787 Cubefull numbers with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 16, 64, 81, 216, 256, 625, 729, 864, 1000, 1024, 1296, 1944, 2401, 2744, 3375, 3456, 4000, 4096, 5184, 6561, 7776, 9261, 10000, 10648, 10976, 11664, 13824, 14641, 15625, 16000, 16384, 17496, 17576, 20736, 25000, 28561, 30375, 31104, 35937, 38416, 39304, 40000

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 even 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 A028260.
Complement of A370788 within A036966.
Subsequence of A370785.
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;}

cp's OEIS Frontend

A051904 Minimal exponent in prime factorization of n.

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A337736 The number of cubefull numbers (A036966) between the consecutive cubes n^3 and (n+1)^3.

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

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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)).

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A362986 a(n) = A000203(A036966(n)), the sum of divisors of the n-th cubefull number A036966(n).

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A376363 The number of distinct prime factors of the cubefull numbers.

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

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