A362976 -id:A362976 - 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).

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)

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

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)

cp's OEIS Frontend

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

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

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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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