A336175 -id:A336175 - cp's OEIS frontend

A338387 Numbers k such that there are no biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

1, 3, 4, 6, 7, 9, 12, 13, 15, 16, 17, 20, 21, 23, 24, 26, 27, 28, 29, 30, 32, 34, 35, 38, 39, 40, 41, 43, 44, 45, 47, 49, 50, 54, 56, 60, 61, 62, 63, 64, 66, 68, 69, 71, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 89, 90, 91, 95, 97, 99, 100, 101, 105, 106, 107

Offset: 1

Amiram Eldar, Oct 23 2020

nonn

Positions of 0's in A338326.

The asymptotic density of this sequence is 0.623265038... (Dehkordi, 1998).

			1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not biquadratefree powerful.
2 is not a term since there is a biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and (2+1)^2 = 9.

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.

Cf. A336175, A338325, A338326, A338327, A338388, A338389, A338390, A338391, A338392.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[100], !AnyTrue[Range[#^2 + 1, (# + 1)^2 - 1], bqfpowQ] &]

A336176 Numbers k such that there is a single powerful number between k^2 and (k+1)^2.

Original entry on oeis.org

2, 8, 10, 15, 16, 18, 19, 20, 28, 29, 37, 39, 41, 42, 45, 48, 50, 51, 52, 53, 56, 57, 59, 63, 65, 74, 76, 77, 78, 79, 83, 84, 87, 89, 90, 92, 94, 100, 101, 102, 107, 113, 114, 115, 116, 117, 118, 119, 121, 122, 126, 127, 130, 134, 138, 141, 144, 146, 147, 148

Offset: 1

Amiram Eldar, Jul 10 2020

nonn

Positions of 1's in A119241.
Shiu (1980) proved that this sequence has an asymptotic density 0.3955... A more accurate calculation using his formula gives 0.3955652153962362...
1 is the most common value of A119241.

			2 is a term since there is a single powerful number, 8 = 2^3, between 2^2 = 4 and (2+1)^2 = 9.

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

Cf. A001694, A119241, A119242, A336175, A336177, A336178.

powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[150], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?powQ] == 1 &]

A336177 Numbers k such that there are exactly two powerful numbers between k^2 and (k+1)^2.

Original entry on oeis.org

5, 11, 14, 22, 25, 33, 44, 46, 55, 58, 62, 70, 72, 73, 82, 88, 96, 98, 103, 104, 109, 110, 111, 124, 129, 135, 155, 156, 158, 164, 172, 176, 178, 181, 187, 197, 203, 206, 207, 209, 212, 218, 240, 243, 248, 249, 254, 257, 259, 268, 277, 279, 281, 285, 288, 291

Offset: 1

Amiram Eldar, Jul 10 2020

nonn

Positions of 2's in A119241.

Shiu (1980) proved that this sequence has an asymptotic density 0.2312... A more accurate calculation using his formula gives 0.231299167354828...

			5 is a term since there are exactly two powerful numbers, 27 = 3^3 and 32 = 2^5 between 5^2 = 25 and (5+1)^2 = 36.

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

Cf. A001694, A119241, A119242, A336175, A336176, A336178.

powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[300], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?powQ] == 2 &]

A336178 Numbers k such that there are exactly three powerful numbers between k^2 and (k+1)^2.

Original entry on oeis.org

31, 36, 67, 93, 132, 140, 145, 161, 166, 189, 192, 220, 223, 265, 280, 290, 296, 311, 316, 322, 364, 384, 407, 468, 537, 576, 592, 602, 623, 639, 644, 656, 659, 661, 670, 690, 722, 769, 771, 793, 828, 883, 888, 890, 896, 950, 961, 981, 984, 987, 992, 995, 1018

Offset: 1

Amiram Eldar, Jul 10 2020

nonn

Positions of 3's in A119241.

Shiu (1980) proved that this sequence has an asymptotic density = 0.0770... A more accurate calculation using his formula gives 0.0770742722233...

			31 is a term since there are exactly three powerful numbers, 968 = 2^3 * 11^2, 972 = 2^2 * 3^5 and 1000 = 2^3 * 5^3 between 31^2 = 961 and (31+1)^2 = 1024.

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

Cf. A001694, A119241, A119242, A336175, A336176, A336177.

powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[1000], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?powQ] == 3 &]

from functools import lru_cache
from math import isqrt
from sympy import mobius, integer_nthroot
def A336178(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
    @lru_cache(maxsize=None)
    def g(x):
        c, l = 0, 0
        j = 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)
        c += squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    def f(x):
        c, a = n+x, 1
        for k in range(1,x+1):
            b = g((k+1)**2)
            if b == a+4:
                c -= 1
            a = b
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 14 2024

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)).

A371187 Numbers k such that there are no cubefull numbers between k^3 and (k+1)^3.

Original entry on oeis.org

1, 11, 16, 23, 72, 84, 140, 144, 197, 208, 223, 252, 286, 296, 300, 306, 313, 353, 477, 500, 502, 525, 528, 620, 671, 694, 721, 734, 737, 751, 785, 802, 827, 858, 900, 913, 916, 976, 1026, 1056, 1059, 1074, 1080, 1143, 1182, 1197, 1230, 1268, 1281, 1284, 1324

Offset: 1

Amiram Eldar, Mar 14 2024

nonn

Positions of 0's in A337736.

This sequence has a positive asymptotic density (Shiu, 1991).

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. Shiu, The distribution of cube-full numbers, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295.
Index entries for sequences related to powerful numbers.

Cf. A036966, A336175, A337736, A371186.

cubQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 2; Select[Range[1000], ! AnyTrue[Range[#^3 + 1, (# + 1)^3 - 1], cubQ] &]
(* or *)
seq[max_] := Module[{cubs = Union[Flatten[Table[i^5*j^4*k^3, {i, 1, Surd[max, 5]}, {j, 1, Surd[max/i^5, 4]}, {k, Surd[max/(i^5*j^4), 3]}]]], s = {}}, Do[If[IntegerQ[Surd[cubs[[k]], 3]], AppendTo[s, k]], {k, 1, Length[cubs]}]; Position[Differences[s], 1] // Flatten]; seq[10^10]

iscub(n) = n == 1 || vecmin(factor(n)[, 2]) >= 3;
is(n) = for(k = n^3+1, (n+1)^3-1, if(iscub(k), return(0))); 1;

1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not cubefull.

cp's OEIS Frontend

A338387 Numbers k such that there are no biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

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A336176 Numbers k such that there is a single powerful number between k^2 and (k+1)^2.

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A336177 Numbers k such that there are exactly two powerful numbers between k^2 and (k+1)^2.

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A336178 Numbers k such that there are exactly three powerful numbers between k^2 and (k+1)^2.

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

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A371187 Numbers k such that there are no cubefull numbers between k^3 and (k+1)^3.

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