A337736 -id:A337736 - cp's OEIS frontend

A362974 Decimal expansion of Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)).

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).

A338326 The number of biquadratefree powerful numbers (A338325) between the consecutive squares n^2 and (n+1)^2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 22 2020

nonn

Dehkordi (1998) proved that for each k>=0 the sequence of numbers m such that a(m) = k has a positive asymptotic density.

			a(2) = 1 since there is one biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and 3^2 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A119241, A337736, A338325.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3 }, #] &]; a[n_] := Count[Range[n^2 + 1, (n + 1)^2 - 1], _?bqfpowQ]; Array[a, 100]

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.

A337737 Least number k such that there are exactly n cubefull numbers between k^3 and (k+1)^3.

Original entry on oeis.org

1, 2, 6, 15, 12, 25, 43, 73, 480, 1981, 3205, 9038, 16099, 376340, 211318, 2461230, 2253517, 16907618, 106308537, 312911063

Offset: 0

Amiram Eldar, Sep 17 2020

a(n) = least k such that A337736(k) = n.

Shiu (1991) proved that infinitely many values of k exist for every n. Therefore, this sequence is infinite.

			a(0) = 1 since there are no cubefull numbers between 1^3 = 1 and 2^3 = 8.
a(1) = 2 since there is one cubefull number, 16 = 2^4, between 2^3 = 8 and 3^3 = 27.
a(2) = 6 since there are 2 cubefull numbers, 243 = 3^5 and 256 = 2^8, between 6^3 = 216 and 7^3 = 343.

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, A119242, A337736.

cubQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 2; f[n_] := Count[Range[n^3 + 1, (n + 1)^3 - 1], _?cubQ]; mx = 8; s = Table[0,{mx}]; c = 0; n = 1; While[c < mx, i = f[n] + 1; If[i <= mx && s[[i]] == 0, c++; s[[i]] = n]; n++] ;s

from math import gcd
from sympy import integer_nthroot, factorint
def A337737(n):
    if n == 0: return 1
    a, k = 0, 1
    while True:
        m, c = k**3, 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]
        if c-a-1 == n:
            return k-1
        k += 1
        a = c # Chai Wah Wu, Apr 23 2025

a(12)-a(16) from David A. Corneth, Sep 18 2020

a(17)-a(19) from Bert Dobbelaere, Sep 19 2020

A371187 Numbers k such that there are no cubefull numbers between k^3 and (k+1)^3.

Original entry on oeis.org

1, 11, 16, 23, 72, 84, 140, 144, 197, 208, 223, 252, 286, 296, 300, 306, 313, 353, 477, 500, 502, 525, 528, 620, 671, 694, 721, 734, 737, 751, 785, 802, 827, 858, 900, 913, 916, 976, 1026, 1056, 1059, 1074, 1080, 1143, 1182, 1197, 1230, 1268, 1281, 1284, 1324

Offset: 1

Amiram Eldar, Mar 14 2024

nonn

Positions of 0's in A337736.

This sequence has a positive asymptotic density (Shiu, 1991).

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.
Index entries for sequences related to powerful numbers.

Cf. A036966, A336175, A337736, A371186.

cubQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 2; Select[Range[1000], ! AnyTrue[Range[#^3 + 1, (# + 1)^3 - 1], cubQ] &]
(* 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]}]; Position[Differences[s], 1] // Flatten]; seq[10^10]

iscub(n) = n == 1 || vecmin(factor(n)[, 2]) >= 3;
is(n) = for(k = n^3+1, (n+1)^3-1, if(iscub(k), return(0))); 1;

1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not cubefull.

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A362974 Decimal expansion of Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)).

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A338326 The number of biquadratefree powerful numbers (A338325) between the consecutive squares n^2 and (n+1)^2.

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

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A337737 Least number k such that there are exactly n cubefull numbers between k^3 and (k+1)^3.

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A371187 Numbers k such that there are no cubefull numbers between k^3 and (k+1)^3.

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