A147533 -id:A147533 - cp's OEIS frontend

A358040 a(n) is the number of divisors of the n-th cubefree number.

1, 2, 2, 3, 2, 4, 2, 3, 4, 2, 6, 2, 4, 4, 2, 6, 2, 6, 4, 4, 2, 3, 4, 6, 2, 8, 2, 4, 4, 4, 9, 2, 4, 4, 2, 8, 2, 6, 6, 4, 2, 3, 6, 4, 6, 2, 4, 4, 4, 2, 12, 2, 4, 6, 4, 8, 2, 6, 4, 8, 2, 2, 4, 6, 6, 4, 8, 2, 4, 2, 12, 4, 4, 4, 2, 12, 4, 6, 4, 4, 4, 2, 6, 6, 9, 2

Offset: 1

Amiram Eldar, Oct 29 2022

nonn

The analogous sequence with squarefree numbers is A072048.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zhu Weiyi, On the cube free number sequences, Smarandache Notions J., Vol. 14 (2004), pp. 199-202.

Cf. A000005, A001620 (gamma), A004709, A072048, A073002 (-zeta'(2)), A147533 (2*gamma-1), A358039.

DivisorSigma[0, Select[Range[100], Max[FactorInteger[#][[;;, 2]]] < 3 &]]

from sympy import mobius, integer_nthroot, divisor_count
def A358040(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_count(m) # Chai Wah Wu, Aug 06 2024

a(n) = A000005(A004709(n)).

Sum_{k=1..n} a(k) = (36*c_1/Pi^4) * n * (log(n) + (2*gamma - 1) - 24*zeta'(2)/Pi^2 - 4*c_2) + O(n^(1/2 + eps)), where c_1 = Product_{p prime} ((p^2+2*p+3)/(p+1)^2) = 1.58095136661854869148023... and c_2 = Sum_{p prime} p*log(p)/((p+1)*(p^2+2*p+3)) = 0.229224... (Weiyi, 2004).

A155739 Decimal expansion of the Euler-Mascheroni constant divided by 2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 26 2009

			0.288607832450766430303256045041201215521079667969961799402...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Ovidiu Furdui, Problem 1870, Mathematics Magazine, Vol. 84, No. 2 (2011), p. 151; A double zeta sum, Solution to Problem 1870 by Joel Schlosberg, ibid., Vol. 85, No. 2 (2012), p. 156.

Cf. A001620, A147533.

R:= RealField(100); EulerGamma(R)/2; // G. C. Greubel, Aug 31 2018

Maple
```
evalf(gamma/2) ;
```

RealDigits[EulerGamma/2 , 10, 100][[1]] (* G. C. Greubel, Aug 31 2018 *)

default(realprecision, 100); Euler/2 \\ G. C. Greubel, Aug 31 2018

Equals A001620/2 = (1 + A147533)/4.

Equals Sum_{k,m>=1} k*(zeta(k+m)-1)/(k+m)^2 (Furdui, 2011). - Amiram Eldar, Jun 09 2022

A351411 Number of divisors of n not of the form p^p, p prime.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 3, 4, 2, 5, 2, 4, 4, 4, 2, 6, 2, 5, 4, 4, 2, 7, 3, 4, 3, 5, 2, 8, 2, 5, 4, 4, 4, 8, 2, 4, 4, 7, 2, 8, 2, 5, 6, 4, 2, 9, 3, 6, 4, 5, 2, 7, 4, 7, 4, 4, 2, 11, 2, 4, 6, 6, 4, 8, 2, 5, 4, 8, 2, 11, 2, 4, 6, 5, 4, 8, 2, 9, 4, 4, 2, 11, 4, 4, 4, 7, 2, 12, 4, 5, 4

Offset: 1

Wesley Ivan Hurt, Feb 10 2022

			a(108) = 10; 2 of the 12 divisors of 108 are of the form p^p (p prime), namely 4 = 2^2 and 27 = 3^3; therefore a(108) = 12-2 = 10.

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

Cf. A000005 (tau), A001221 (omega), A001222 (Omega), A007947 (rad).

Cf. A051674, A129251.

f1[p_, e_] := e + 1; f2[p_, e_] := If[e < p, 0, 1]; a[1] = 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Plus @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)

a(n) = {my(f = factor(n)); vecprod(apply(x -> x+1, f[, 2])) - sum(i = 1, #f~, f[i, 2] >= f[i, 1]); } \\ Amiram Eldar, Oct 01 2023

a(n) = tau(n) - Sum_{d|n} [rad(d) = Omega(d)*[omega(d) = 1]], where [ ] is the Iverson bracket.
a(n) = A000005(n) - A129251(n).
Sum_{k=1..n} a(k) ~ n * (log(n) + c), where c = A147533 - A094289 = -0.1329269215... . Amiram Eldar, Oct 01 2023

A386713 Decimal expansion of Integral_{x=0..1} {1/x}^2 * {1/(1-x)}^2 dx, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 31 2025

			0.04264560603125049181658953091533139472254244534257...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.11, page 101.

Ovidiu Furdui, A class of fractional part integrals and zeta function values, Integral Transforms and Special Functions, Vol. 24, No. 6 (2013), pp. 485-490.

Cf. A001620 (gamma), A061444, A345208.

Cf. A147533 (m=1), this constant (m=2), A386714 (m=3).

RealDigits[4*Log[2*Pi] - 4*EulerGamma - 5, 10, 120, -1][[1]]

PARI
```
4*log(2*Pi) - 4*Euler - 5
```

Equals 4*log(2*pi) - 4*gamma - 5.
Equals 4*A345208 - 1.
In general, for m >= 2, Integral_{x=0..1} {1/x}^m * {1/(1-x)}^m dx = 2 * (Sum_{j=2..m-1} (-1)^(m+j-1) * (zeta(j)-1)) + (-1)^m - (2*m) * Sum_{k>=0} (zeta(2*k+m) - zeta(2*k+m+1))/(k+m) (note that the first sum vanishes when m = 2).

A386714 Decimal expansion of Integral_{x=0..1} {1/x}^3 * {1/(1-x)}^3 dx, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 31 2025

			0.01461170261653269568427203547387356507606811502683...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.11, page 101.

Ovidiu Furdui, A class of fractional part integrals and zeta function values, Integral Transforms and Special Functions, Vol. 24, No. 6 (2013), pp. 485-490.

Cf. A001620 (gamma), A013661, A061444, A074962 (A), A225746.

Cf. A147533 (m=1), A386713 (m=2), this constant (m=3).

RealDigits[-Zeta[2] + 3*EulerGamma + 36*Log[Glaisher] - 6*Log[2*Pi] + 2, 10, 120, -1][[1]]

-zeta(2) + 3*Euler + 36*(1/12-zeta'(-1)) - 6*log(2*Pi) + 2

Equals -zeta(2) + 3*gamma + 36*log(A) - 6*log(2*Pi) + 2, where gamma is Euler's constant and A is the Glaisher-Kinkelin constant.

In general, for m >= 2, Integral_{x=0..1} {1/x}^m * {1/(1-x)}^m dx = 2 * (Sum_{j=2..m-1} (-1)^(m+j-1) * (zeta(j)-1)) + (-1)^m - (2*m) * Sum_{k>=0} (zeta(2*k+m) - zeta(2*k+m+1))/(k+m) (note that the first sum vanishes when m = 2).

A335007 Decimal expansion of 2*(gamma - zeta'(2)/zeta(2)) - 1, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 19 2020

			1.2943533159921313340127529002042648668912832334937...

Eckford Cohen, The number of unitary divisors of an integer, The American Mathematical Monthly, Vol. 67, No. 9 (1960), pp. 879-880.

Cf. A001620 (gamma), A013661 (zeta(2)), A034444, A064608, A073002 (-zeta'(2)), A147533, A335006.

RealDigits[2*EulerGamma - 2*Zeta'[2]/Zeta[2] - 1, 10, 100][[1]]

2*Euler - 2*zeta'(2)/zeta(2) - 1 \\ Michel Marcus, May 19 2020

Equals lim_{k->oo} ((zeta(2)/k)*A064608(k) - log(k)) where A064608 is the partial sums of the number of unitary divisors (A034444).

Equals 2*A001620 + 2*A073002/A013661 - 1 = 2*A335006 - 1.

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A358040 a(n) is the number of divisors of the n-th cubefree number.

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A155739 Decimal expansion of the Euler-Mascheroni constant divided by 2.

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A351411 Number of divisors of n not of the form p^p, p prime.

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A386713 Decimal expansion of Integral_{x=0..1} {1/x}^2 * {1/(1-x)}^2 dx, where {} denotes fractional part.

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A386714 Decimal expansion of Integral_{x=0..1} {1/x}^3 * {1/(1-x)}^3 dx, where {} denotes fractional part.

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A335007 Decimal expansion of 2*(gamma - zeta'(2)/zeta(2)) - 1, where gamma is the Euler-Mascheroni constant.

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