A244000 -id:A244000 - 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).

A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

Original entry on oeis.org

1, 4, 8, 9, 25, 27, 36, 49, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 289, 343, 361, 392, 441, 484, 500, 529, 675, 676, 841, 900, 961, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1331, 1352, 1369, 1372, 1444, 1521, 1681, 1764, 1800, 1849, 2116, 2197, 2209

Offset: 1

Amiram Eldar, Oct 22 2020

nonn

Equivalently, numbers k such that if a prime p divides k then p^2 divides k but p^4 does not divide k.
Each term has a unique representation as a^2 * b^3, where a and b are coprime squarefree numbers.
Dehkordi (1998) refers to these numbers as "2-full and 4-free numbers".

			4 = 2^2 is a term since the exponent of its only prime factor is 2.
72 = 2^3 * 3^2 is a terms since the exponents of the primes in its prime factorization are 2 and 3.

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.
Eric Weisstein's World of Mathematics, Biquadratefree.
Eric Weisstein's World of Mathematics, Powerful Number.

Intersection of A001694 and A046100.
Subsequences: A062503, A062838.
Cf. A005117, A090699, A244000, A330595.

Select[Range[2500], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], MemberQ[{2, 3}, #1] &] &]

The number of terms not exceeding x is asymptotically (zeta(3/2)/zeta(3)) * J_2(1/2) * x^(1/2) + (zeta(2/3)/zeta(2)) * J_2(1/3) * x^(1/3), where J_2(s) = Product_{p prime} (1 - p^(-4*s) - p^(-5*s) - p^(-6*s) + p^(-7*s) + p^(-8*s)) (Dehkordi, 1998).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) = 1.748932... (A330595).

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)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

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)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI