A336177 -id:A336177 - cp's OEIS frontend

A336175 Numbers k such that there are no powerful numbers between k^2 and (k+1)^2.

1, 3, 4, 6, 7, 9, 12, 13, 17, 21, 23, 24, 26, 27, 30, 32, 34, 35, 38, 40, 43, 47, 49, 54, 60, 61, 64, 66, 68, 69, 71, 75, 80, 81, 85, 86, 91, 95, 97, 99, 105, 106, 108, 112, 120, 123, 125, 128, 131, 133, 136, 137, 139, 142, 143, 151, 153, 154, 159, 160, 162, 163

Offset: 1

Amiram Eldar, Jul 10 2020

nonn

Positions of 0's in A119241.

Shiu (1980) proved that this sequence has an asymptotic density 0.2759... A more accurate calculation using his formula gives 0.275965511407718981...

			1 is a term since the two numbers between 1^2 = 1 and (1+1)^2 = 4, 2 and 3, are not powerful.
2 is not a term since there is a 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, A336176, A336177, A336178.

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

is(n)=forfactored(k=n^2+1,n^2+2*n, if(ispowerful(k), return(0))); 1 \\ Charles R Greathouse IV, Oct 31 2022

from functools import lru_cache
from math import isqrt
from sympy import mobius, integer_nthroot
def A336175(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+1:
                c -= 1
            a = b
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 14 2024

a(n) ~ k*n, where k = 3.623641211175... is the inverse of the density, see Shiu link. - Charles R Greathouse IV, Oct 31 2022

A338389 Numbers k such that there are exactly two biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

14, 31, 67, 72, 82, 93, 98, 110, 132, 140, 156, 172, 189, 192, 223, 240, 257, 281, 285, 322, 347, 368, 379, 407, 410, 414, 426, 441, 455, 468, 472, 481, 488, 514, 515, 517, 524, 525, 537, 551, 555, 574, 579, 602, 613, 664, 680, 693, 702, 703, 737, 743, 749, 755

Offset: 1

Amiram Eldar, Oct 23 2020

nonn

Positions of 2's in A338326.

The asymptotic density of this sequence is 0.058757863... (Dehkordi, 1998).

			14 is a term since there are exactly two biquadratefree powerful numbers, 200 = 2*3 * 5^2 and 216 = 2^3 * 3^3, between 14^2 = 196 and (14+1)^2 = 225.

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. A336177, A338325, A338326, A338327, A338387, A338388, A338390, A338391, A338392.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3}, #] &]; Select[Range[800], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?bqfpowQ] == 2 &]

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 &]

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

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A336175 Numbers k such that there are no powerful numbers between k^2 and (k+1)^2.

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A338389 Numbers k such that there are exactly two 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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A336178 Numbers k such that there are exactly three powerful numbers between k^2 and (k+1)^2.

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Mathematica

Python