A090699 -id:A090699 - cp's OEIS frontend

A376361 The number of distinct prime factors of the powerful numbers.

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

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, Distribution of omega(n) over h-free and h-full numbers, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.2.
Index entries for sequences related to powerful numbers.

Cf. A001221, A001694, A072047, A077761, A090699, A376362, A376363, A376365.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 1 &], Length[e], Nothing]]; Array[f, 3500]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] == 1, is = 0; break)); if(is, print1(#e, ", ")));}

a(n) = A001221(A001694(n)).

Sum_{A001694(k) <= x} a(k) = c * sqrt(x) * (log(log(x)) + B - log(2) + L(2, 3) - L(2, 4)) + O(sqrt(x)/log(x)), where c = zeta(3/2)/zeta(3) (A090699), B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), L(2, 3) = 1.07848461669337535407..., and L(2, 4) = 0.57937575954505652569... (Das et al., 2024).

A376362 The number of unitary divisors that are squares of primes applied to the powerful numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, On the number of prime factors with a given multiplicity over h-free and h-full numbers, Journal of Number Theory, Vol. 267 (2025), pp. 176-201; arXiv preprint, arXiv:2409.11275 [math.NT], 2024. See Theorem 1.3.
Index entries for sequences related to powerful numbers.

Cf. A001694, A077761, A090699, A369427, A376361, A376364, A376366.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 1 &], Count[e, 2], Nothing]]; Array[f, 3500]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] == 1, is = 0; break)); if(is, print1(#select(x -> x == 2, e), ", ")));}

a(n) = A369427(A001694(n)).

Sum_{A001694(k) <= x} a(k) = c * sqrt(x) * (log(log(x)) + B - log(2) - L(2, 4)) + O(sqrt(x)/log(x)), where c = zeta(3/2)/zeta(3) (A090699), B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), and L(2, 4) = 0.57937575954505652569... (Das et al., 2025).

A244000 Decimal expansion of the Bateman-Grosswald constant zeta(2/3)/zeta(2), a constant (negated) arising in the asymptotic evaluation of the number of square-full numbers (also called "powerful" numbers).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 17 2014

			-1.4879506635322726315987491125787134987961...

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

Wikipedia, Powerful number

Cf. A001694, A033150, A059956 (analog for squarefree integers), A090699.

RealDigits[ Zeta[2/3] / Zeta[2], 10, 101] // First

A361936 Indices of the squares in the sequence of powerful numbers (A001694).

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 11, 13, 14, 16, 19, 20, 21, 24, 26, 28, 29, 31, 33, 35, 36, 39, 40, 41, 44, 45, 46, 48, 50, 51, 55, 56, 59, 60, 61, 65, 67, 68, 70, 71, 73, 75, 76, 79, 81, 84, 85, 87, 88, 90, 92, 94, 96, 97, 100, 102, 104, 107, 109, 110, 111, 114, 116, 117, 119, 120

Offset: 1

Amiram Eldar, Mar 31 2023

nonn

Equivalently, the number of powerful numbers that do not exceed n^2.
The asymptotic density of this sequence is zeta(3)/zeta(3/2) = 1/A090699 = 0.460139... .
If k is a term of A336175 then a(k) and a(k+1) are consecutive integers, i.e., a(k+1) = a(k) + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), Vol. 7, No. 2 (1935), pp. 95-102.
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77, No. 8 (1970), pp. 848-852.

Cf. A000290, A001694, A090699, A119241, A217038, A336175.

Position[Select[Range[5000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(IntegerQ[Sqrt[#]] &)] // Flatten

lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++); if(issquare(k), print1(c, ", "))); }

from math import isqrt
from sympy import integer_nthroot, factorint
def A361936(n):
    m = n**2
    return int(sum(isqrt(m//k**3) for k in range(1, integer_nthroot(m, 3)[0]+1) if all(d<=1 for d in factorint(k).values()))) # Chai Wah Wu, Sep 10 2024

a(n) = A217038(n^2).
a(n+1) - a(n) = A119241(n) + 1.
a(n) = (zeta(3/2)/zeta(3)) * n + O(n^(2/3)).

A370786 Powerful numbers with an odd number of prime factors (counted with multiplicity).

Original entry on oeis.org

8, 27, 32, 72, 108, 125, 128, 200, 243, 288, 343, 392, 432, 500, 512, 648, 675, 800, 968, 972, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1728, 1800, 2000, 2048, 2187, 2197, 2312, 2592, 2700, 2888, 3087, 3125, 3200, 3267, 3528, 3872, 3888, 4232, 4500, 4563, 4608

Offset: 1

Amiram Eldar, Mar 02 2024

Jakimczuk (2024) proved:
The number of terms that do not exceed x is N(x) = c * sqrt(x) + o(sqrt(x)) where c = (zeta(3/2)/zeta(3) - 1/zeta(3/2))/2 = 0.895230... .
The relative asymptotic density of this sequence within the powerful numbers is (1 - zeta(3)/(zeta(3/2)^2))/2 = 0.411930... .
In general, the relative asymptotic density of the s-full numbers (numbers whose exponents in their prime factorization are all >= s) with an odd number of prime factors (counted with multiplicity) within the s-full numbers is smaller than 1/2 when s is odd.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Intersection of A001694 and A026424.
Complement of A370785 within A001694.
A370788 is a subsequence.
Cf. A002117, A078434, A090699.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, # > 1 &] && OddQ[Total[e]]]; Select[Range[2500], q]

is(n) = {my(e = factor(n)[, 2]); n > 1 && vecmin(e) > 1 && vecsum(e)%2;}

A084371 Squarefree kernels of powerful numbers (A001694).

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 3, 2, 6, 7, 2, 6, 3, 10, 6, 11, 5, 2, 6, 13, 14, 10, 6, 15, 3, 2, 6, 17, 6, 7, 19, 14, 10, 6, 21, 22, 10, 2, 23, 6, 5, 6, 15, 26, 3, 14, 10, 29, 6, 30, 31, 22, 6, 10, 2, 33, 15, 6, 34, 35, 6, 21, 11, 26, 37, 14, 38, 39, 14, 10, 41, 6, 42, 30, 43, 22, 6, 10

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

			A001694(11) = 64 = 2^6 -> a(11) = 2,
A001694(12) = 72 = 2^3 * 3^2 -> a(12) = 2*3 = 6,
A001694(13) = 81 = 3^4 -> a(13) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, The kernel of powerful numbers, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, Remark 2.5, p. 729.

Cf. A001694, A007947, A090699.

s = {1}; Do[f = FactorInteger[n]; If[Min @ f[[;;, 2]] > 1, AppendTo[s, Times @@ f[[;;, 1]]]], {n, 2, 10^4}]; s (* Amiram Eldar, Aug 22 2019 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = apply(x->rad(x), select(x->ispowerful(x), [1..nn])); \\ Michel Marcus, Aug 22 2019

from math import prod, isqrt
from sympy import mobius, integer_nthroot, primefactors
def A084371(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, l, j = n+x-squarefreepi(integer_nthroot(x,3)[0]), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return prod(primefactors(bisection(f,n,n))) # Chai Wah Wu, Sep 13 2024

a(n) = A007947(A001694(n)).
From Amiram Eldar, May 13 2023: (Start)
Sum_{A001694(k) < x} a(k) = (1/2) * x + o(x) (Jakimczuk, 2017). [corrected Sep 21 2024]
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/zeta(3/2))^2/2 = 0.1058641473... . (End)

A174172 Partials sums of A001694.

Original entry on oeis.org

1, 5, 13, 22, 38, 63, 90, 122, 158, 207, 271, 343, 424, 524, 632, 753, 878, 1006, 1150, 1319, 1515, 1715, 1931, 2156, 2399, 2655, 2943, 3232, 3556, 3899, 4260, 4652, 5052, 5484, 5925, 6409, 6909, 7421, 7950, 8526, 9151, 9799, 10474, 11150, 11879, 12663

Offset: 1

Jonathan Vos Post, Mar 10 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, The kernel of powerful numbers, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, Theorem 2.7, p. 729.

Cf. A001694, A090699.

Accumulate @ Select[Range[1000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &] (* Amiram Eldar, Jan 30 2023 *)

lista(kmax) = {my(s = 0); for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, s += k; print1(s, ", ")));} \\ Amiram Eldar, May 13 2023

a(n) = Sum_{i=1..n} A001694(i).
a(n) ~ (zeta(3)^2/(3*zeta(3/2)^2)) * n^3. - Amiram Eldar, Jan 30 2023
a(n) = c * A001694(n)^(3/2) + o(A001694(n)^(3/2)), where c = zeta(3/2)/(3*zeta(3)) = 0.7244181041... (Jakimczuk, 2017). - Amiram Eldar, May 13 2023

A370785 Powerful numbers with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 216, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 864, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 1944, 2025, 2116, 2209

Offset: 1

Amiram Eldar, Mar 02 2024

Jakimczuk (2024) proved:
The number of terms that do not exceed x is N(x) = c * sqrt(x) + o(sqrt(x)) where c = (zeta(3/2)/zeta(3) + 1/zeta(3/2))/2 = 1.278023... .
The relative asymptotic density of this sequence within the powerful numbers is (1 + zeta(3)/(zeta(3/2)^2))/2 = 0.588069... .
In general, the relative asymptotic density of the s-full numbers (numbers whose exponents in their prime factorization are all >= s) with an even number of prime factors (counted with multiplicity) within the s-full numbers is larger than 1/2 when s is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Intersection of A001694 and A028260.
Complement of A370786 within A001694.
A370787 is a subsequence.
Cf. A002117, A078434, A090699.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, n == 1 || AllTrue[e, # > 1 &] && EvenQ[Total[e]]]; Select[Range[2500], q]

is(n) = {my(e = factor(n)[, 2]); n == 1 || (vecmin(e) > 1 && !(vecsum(e)%2));}

A380266 a(n) is the numerator of the mean value of A051904(k) at the range k = 1..n.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 5, 4, 13, 14, 5, 16, 17, 6, 11, 23, 4, 25, 13, 9, 14, 29, 5, 32, 33, 4, 37, 38, 13, 40, 45, 46, 47, 48, 25, 51, 26, 53, 27, 55, 4, 57, 29, 59, 30, 61, 31, 64, 13, 22, 67, 68, 23, 14, 71, 24, 73, 74, 5, 76, 77, 26, 21, 17, 43, 87, 22, 89, 9

Offset: 1

Amiram Eldar, Jan 18 2025

			Fractions begin with 0, 1/2, 2/3, 1, 1, 1, 1, 5/4, 4/3, 13/10, 14/11, 5/4, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant.
Wikipedia, Niven's constant.

Cf. A051904, A090699, A380267 (denominators).

f[n_] := Min[FactorInteger[n][[;;, 2]]]; f[1] = 0; With[{m = 100}, Numerator[Accumulate[Array[f, m]] / Range[m]]]

lista(nmax) = {my(s = 0); print1(0, ", "); for(n = 2, nmax, s += vecmin(factor(n)[,2]);  print1(numerator(s/n), ", "));}

a(n) = numerator((Sum_{k=1..n} A051904(k))/n).

a(n)/A380267(n) = 1 + c/sqrt(n) + o(1/sqrt(n)), where c = zeta(3/2)/zeta(3) (A090699).

A217038 Number of powerful numbers < n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12

Offset: 1

Jayanta Basu, Apr 07 2013

nonn

Powerful numbers are given by A001694.

			a(10)=4 since there are exactly 4 powerful numbers (1,4,8,9) less than 10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), Vol. 7, No. 2 (1935), pp. 95-102.
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77, No. 8 (1970), pp. 848-852.

Cf. A001597, A001694, A076411, A090699.

Partial sums of A112526.

PowQ[n_] := Cases[FactorInteger[n], {p_, 1} -> p] == {}; Q[n_] := Length[Join[{1}, Select[Range[n - 1], PowQ[#] &]]] ; Join[{0}, Table[Q[n], {n, 2, 100}]]

g(n,fe=factor(n)[,2])=prod(i=1,#fe, (fe[i]+2)\2 - (fe[i]+2)\3)
a(n)=my(v=List(),t); n--; for(m=2,sqrtnint(n,6), for(y=1,sqrtnint(n\m^6,3), t=(m^2*y)^3; for(x=1,sqrtint(n\t), listput(v,t*x^2)))); v=Set(v); sum(y=1,sqrtnint(n,3), sqrtint(n\y^3))-sum(i=1,#v, g(v[i])-1) \\ Charles R Greathouse IV, Jul 31 2017

first(n)=my(v=vector(n),s=1); if(n>1, v[2]=1); forfactored(k=2,n-1, if(vecmin(k[2][,2])>1, s++); v[k[1]+1]=s); v \\ Charles R Greathouse IV, Jul 31 2017

a(n)=my(s); n--; forsquarefree(k=1,sqrtnint(n,3), s+=sqrtint(n\k[1]^3)); s \\ Charles R Greathouse IV, Dec 12 2022

from math import isqrt
from sympy import mobius, integer_nthroot
def A217038(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    c, l = 0, 0
    j = isqrt(n-1)
    while j>1:
        k2 = integer_nthroot((n-1)//j**2,3)[0]+1
        w = squarefreepi(k2-1)
        c += j*(w-l)
        l, j = w, isqrt((n-1)//k2**3)
    c += squarefreepi(integer_nthroot(n-1,3)[0])-l
    return c # Chai Wah Wu, Sep 12 2024

a(n) = (zeta(3/2)/zeta(3)) * sqrt(n) + O(n^(1/3)) (Erdős and Szekeres, 1935; Golomb, 1970). - Amiram Eldar, Apr 06 2023

cp's OEIS Frontend

A376361 The number of distinct prime factors of the powerful numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A376362 The number of unitary divisors that are squares of primes applied to the powerful numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A244000 Decimal expansion of the Bateman-Grosswald constant zeta(2/3)/zeta(2), a constant (negated) arising in the asymptotic evaluation of the number of square-full numbers (also called "powerful" numbers).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

A361936 Indices of the squares in the sequence of powerful numbers (A001694).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A370786 Powerful numbers with an odd number of prime factors (counted with multiplicity).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A084371 Squarefree kernels of powerful numbers (A001694).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A174172 Partials sums of A001694.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A370785 Powerful numbers with an even number of prime factors (counted with multiplicity).

Views

Author

Keywords

Comments