A090699 -id:A090699 - cp's OEIS frontend

A242977 Decimal expansion of Sum_{k>1} 1/(k(k-1)zeta(k)), a constant related to Niven's constant.

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

Offset: 0

Jean-François Alcover, May 28 2014

The asymptotic mean of the reciprocals of the maximal exponent in prime factorization of the positive integers. - Amiram Eldar, Dec 15 2022

			0.766944490521088241652417923...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6 Niven's constant, p. 113.

Wolfgang Schwarz and Jürgen Spilker, A remark on some special arithmetical functions, in: E. Laurincikas , E. Manstavicius and V. Stakenas (eds.), Analytic and Probabilistic Methods in Number Theory, Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996, New Trends in Probability and Statistics, Vol. 4, VSP BV & TEV Ltd. (1997), pp. 221-245.
D. Suryanarayana and R. Sita Rama Chandra Rao, On the maximum and minimum exponents in factoring integers, Archiv der Mathematik, Vol. 28, No. 1 (1977), pp. 261-269.
Eric Weisstein's World of Mathematics, Niven's Constant.

Cf. A033150, A051903, A072102, A090699.

digits = 98; m0 = 100; dm = 100; Clear[f]; f[m_] := f[m] = NSum[1/(k*(k - 1)*Zeta[k]), {k, 2, m}, WorkingPrecision -> digits + 10, NSumTerms -> m] + 1/m; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], Print["m = ", m ]; m = m + dm]; RealDigits[f[m], 10, digits] // First

sumpos(k = 2, 1/(k*(k-1)*zeta(k))) \\ Amiram Eldar, Dec 15 2022

Equals lim_{n->oo} (1/n) * Sum_{k=2..n} 1/A051903(k). - Amiram Eldar, Oct 16 2020

Equals 1 + Sum_{k>=2} (1/zeta(k)-1)/(k*(k-1)). - Amiram Eldar, Dec 15 2022

A360729 a(n) is the number of prime factors of the n-th powerful number (counted with repetition).

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 5, 4, 4, 5, 2, 3, 7, 6, 2, 4, 5, 6, 4, 5, 8, 7, 2, 6, 3, 2, 5, 6, 7, 4, 4, 5, 9, 2, 8, 4, 7, 5, 4, 6, 6, 7, 2, 8, 6, 2, 5, 7, 6, 10, 4, 5, 9, 4, 4, 8, 5, 3, 5, 2, 5, 4, 4, 7, 8, 2, 9, 6, 7, 2, 6, 8, 7, 6, 11, 4, 7, 3, 2, 10, 5

Offset: 1

Amiram Eldar, Feb 18 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

Cf. A001222, A001694, A083342, A090699.

Similar sequences: A072047, A076399.

PrimeOmega[Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]

apply(bigomega, select(ispowerful, [1..3000]))

a(n) = A001222(A001694(n)).

Sum_{A001694(k) < x} a(k) = (2*zeta(3/2)/zeta(3))*sqrt(x)*log(log(x)) + (2*(B_2 - log(2)) + Sum_{p prime} (3/((p^(3/2)+1))))*(zeta(3/2)/zeta(3))*sqrt(x) + O(sqrt(x)/sqrt(log(x))), where B_2 = A083342 (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

A371185 Indices of the cubefull numbers in the sequence of powerful numbers.

Original entry on oeis.org

1, 3, 5, 7, 8, 11, 13, 17, 18, 23, 25, 26, 30, 34, 38, 41, 42, 45, 49, 54, 55, 61, 63, 72, 77, 78, 80, 82, 83, 87, 89, 93, 99, 105, 106, 113, 115, 116, 127, 128, 130, 137, 140, 148, 151, 153, 161, 164, 166, 179, 185, 186, 188, 192, 196, 201, 206, 221, 227, 234

Offset: 1

Amiram Eldar, Mar 14 2024

			The first 5 powerful numbers are 1, 4, 8, 9, and 16. The 1st, 3rd, and 5th, 1, 8, and 16, are also cubefull numbers. Therefore, the first 3 terms of this sequence are 1, 3, and 5.

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

Cf. A001694, A036966.
Cf. A090699, A362974.
Similar sequences: A361936, A371186.

powQ[n_, emin_] := n == 1 || AllTrue[FactorInteger[n], Last[#] >= emin &]; Position[Select[Range[20000], powQ[#, 2] &], _?(powQ[#1, 3] &), Heads -> False] // Flatten

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

from math import isqrt, gcd
from sympy import mobius, integer_nthroot, factorint
def A371185(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        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
    c, l, m = 0, 0, bisection(f,n,n)
    j = isqrt(m)
    while j>1:
        k2 = integer_nthroot(m//j**2,3)[0]+1
        w = squarefreepi(k2-1)
        c += j*(w-l)
        l, j = w, isqrt(m//k2**3)
    c += squarefreepi(integer_nthroot(m,3)[0])-l
    return c # Chai Wah Wu, Sep 12 2024

A001694(a(n)) = A036966(n).

a(n) ~ c * n^(3/2), where c = A090699 / A362974^(3/2) = 0.216089803749...

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)

A235872 Number of solutions to the equation x^2=0 in the ring of Gaussian integers modulo n.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 9, 2, 1, 4, 1, 2, 1, 16, 1, 18, 1, 4, 1, 2, 1, 8, 25, 2, 9, 4, 1, 2, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 2, 1, 4, 9, 2, 1, 16, 49, 50, 1, 4, 1, 18, 1, 8, 1, 2, 1, 4, 1, 2, 9, 64, 1, 2, 1, 4, 1, 2, 1, 72, 1, 2, 25, 4, 1, 2, 1, 16, 81, 2

Offset: 1

José María Grau Ribas, Apr 03 2014

Numbers of solutions to x^2 == y^2 (mod n), 2*x*y == 0 (mod n). - Andrew Howroyd, Aug 06 2018

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A062803, A090699.

invoG[n_] := invoG[n] = Sum[If[Mod[(x + I y)^2, n] == 0, 1, 0], {x, 0, n - 1}, {y, 0, n - 1}]; Table[invoG[n], {n, 1, 104}]
f[p_, e_] := p^(2*Floor[e/2]); f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2022 *)

a(n)={sum(i=0, n-1, sum(j=0, n-1, (i^2 - j^2)%n == 0 && 2*i*j%n == 0))} \\ Andrew Howroyd, Aug 06 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); p^if(p==2, e, e - e%2))} \\ Andrew Howroyd, Aug 06 2018

Multiplicative with a(2^e) = 2^e, a(p^e) = p^(2*floor(e/2)). - Andrew Howroyd, Aug 06 2018

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (2/21)*(3+sqrt(2))*zeta(3/2)/zeta(3) = 0.91363892007.... - Amiram Eldar, Nov 13 2022

A382790 a(n) is the (2^n)-th powerful number.

Original entry on oeis.org

1, 4, 9, 32, 121, 392, 1352, 5000, 18432, 69192, 265837, 1024144, 3968064, 15523600, 60972500, 240413400, 950612224, 3767130288, 14959246864, 59495990724, 236902199076, 944193944097, 3765996039168, 15029799230264, 60010866324576, 239700225078125, 957712290743329

Offset: 0

Amiram Eldar, Apr 05 2025

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..51 (terms 0..43 from Amiram Eldar)
Index entries for sequences related to powerful numbers.

Cf. A001694, A062762, A090699, A376092.

seq[max_] := Module[{p = Union@ Flatten@ Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]}, p[[2^Range[0, Floor[Log2[Length[p]]]]]]]; seq[10^12]

a(n) = A001694(2^n).

a(n) ~ c * 4^n, where c = (zeta(3)/zeta(3/2))^2 = 1/A090699^2.

A242972 Decimal expansion of a constant related to Niven's constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 28 2014

			0.89289457145126609045700943002242709336...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6 Niven's constant, p. 113.

Eric Weisstein's MathWorld, Niven's Constant

Cf. A033150, A072102, A090699.

digits = 100; k0 = 100; dk = 50; $MaxExtraPrecision = 12*digits; z[n_?NumericQ] := Zeta[Prime[n // Floor]]; Clear[s]; s[k_] := s[k] = NSum[z[n] - 1, {n, 1, k}, WorkingPrecision -> digits + 10, NSumTerms -> 10*digits]*(1 + NSum[Zeta[n] - 1, {n, k + 1, Infinity}, WorkingPrecision -> digits + 10]); s[k0] ; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits] != RealDigits[s[k - dk], 10, digits], Print["k = ", k]; k = k + dk]; RealDigits[s[k], 10, digits] // First

Equals Sum_(p prime) (zeta(p)-1).

Equals Sum_{k>=2} Sum_{p prime} 1/k^p. - Amiram Eldar, Aug 21 2020

cp's OEIS Frontend

A242977 Decimal expansion of Sum_{k>1} 1/(k*(k-1)*zeta(k)), a constant related to Niven's constant.

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A360729 a(n) is the number of prime factors of the n-th powerful number (counted with repetition).

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A371185 Indices of the cubefull numbers in the sequence of powerful numbers.

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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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A235872 Number of solutions to the equation x^2=0 in the ring of Gaussian integers modulo n.

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A382790 a(n) is the (2^n)-th powerful number.

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A242977 Decimal expansion of Sum_{k>1} 1/(k(k-1)zeta(k)), a constant related to Niven's constant.