A119242 -id:A119242 - cp's OEIS frontend

A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.

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

Offset: 1

T. D. Noe, May 09 2006

nonn

Is there an upper bound on the number of powerful numbers between consecutive squares? Pettigrew conjectures that there is no bound. See A119242.

This sequence is unbounded. For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1980). - Amiram Eldar, Jul 10 2020

			a(5) = 2 because the two powerful numbers 27 and 32 are between 25 and 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, A090699, A119242.

Powerful[n_] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Table[Length[Select[Range[k^2+1, k^2+2k], Powerful[ # ]&]], {k,130}]

from math import isqrt
from sympy import integer_nthroot, factorint
def A119241(n):
    def f(x): return int(sum(isqrt(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1) if all(d<=1 for d in factorint(k).values())))
    return f((n+1)**2-1)-f(n**2) # Chai Wah Wu, Sep 10 2024

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3/2)/zeta(3) - 1 = A090699 - 1 = 1.173254... - Amiram Eldar, Oct 24 2020

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

Original entry on oeis.org

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

A338327 a(n) is the least number k such that there are exactly n biquadratefree powerful numbers (A338325) between k^2 and (k+1)^2.

Original entry on oeis.org

1, 2, 14, 36, 234, 3510, 211297, 487425, 20136429

Offset: 0

Amiram Eldar, Oct 22 2020

a(n) is the least k such that A338326(k) = n.
Dehkordi (1998) proved that for each k>=0 the sequence of numbers m such that A338326(m) = k has a positive asymptotic density. Therefore, this sequence is infinite.
a(9) > 10^10. - Bert Dobbelaere, Oct 29 2020

			a(0) = 1 since there are no biquadratefree powerful numbers between 1^2 = 1 and 2^2 = 4.
a(1) = 2 since there is one biquadratefree powerful number, 8 = 2^3, between 2^2 = 4 and 3^2 = 8.
a(2) = 14 since there are 2 biquadratefree powerful numbers, 200 = 2^3 * 5^2 and 216 = 2^3 * 3^3, between 14^2 = 196 and 15^2 = 225.

Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.

Cf. A119242, A337737, A338325, A338326.

bqfpowQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], MemberQ[{2, 3 }, #] &]; f[n_] := Count[Range[n^2 + 1, (n + 1)^2 - 1], _?bqfpowQ]; mx = 5; s = Table[0, {mx}]; c = 0; n = 1; While[c < mx, i = f[n] + 1; If[i <= mx && s[[i]] == 0, c++; s[[i]] = n]; n++]; s

a(8) from Bert Dobbelaere, Oct 29 2020

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

A337737 Least number k such that there are exactly n cubefull numbers between k^3 and (k+1)^3.

Original entry on oeis.org

1, 2, 6, 15, 12, 25, 43, 73, 480, 1981, 3205, 9038, 16099, 376340, 211318, 2461230, 2253517, 16907618, 106308537, 312911063

Offset: 0

Amiram Eldar, Sep 17 2020

a(n) = least k such that A337736(k) = n.

Shiu (1991) proved that infinitely many values of k exist for every n. Therefore, this sequence is infinite.

			a(0) = 1 since there are no cubefull numbers between 1^3 = 1 and 2^3 = 8.
a(1) = 2 since there is one cubefull number, 16 = 2^4, between 2^3 = 8 and 3^3 = 27.
a(2) = 6 since there are 2 cubefull numbers, 243 = 3^5 and 256 = 2^8, between 6^3 = 216 and 7^3 = 343.

P. Shiu, The distribution of cube-full numbers, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295. See section 3, p. 291.

Cf. A000578, A036966, A119242, A337736.

cubQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 2; f[n_] := Count[Range[n^3 + 1, (n + 1)^3 - 1], _?cubQ]; mx = 8; s = Table[0,{mx}]; c = 0; n = 1; While[c < mx, i = f[n] + 1; If[i <= mx && s[[i]] == 0, c++; s[[i]] = n]; n++] ;s

from math import gcd
from sympy import integer_nthroot, factorint
def A337737(n):
    if n == 0: return 1
    a, k = 0, 1
    while True:
        m, c = k**3, 0
        for x in range(1,integer_nthroot(m,5)[0]+1):
            if all(d<=1 for d in factorint(x).values()):
                for y in range(1,integer_nthroot(z:=m//x**5,4)[0]+1):
                    if gcd(x,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        if c-a-1 == n:
            return k-1
        k += 1
        a = c # Chai Wah Wu, Apr 23 2025

a(12)-a(16) from David A. Corneth, Sep 18 2020

a(17)-a(19) from Bert Dobbelaere, Sep 19 2020

cp's OEIS Frontend

A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.

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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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A338327 a(n) is the least number k such that there are exactly n 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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A337737 Least number k such that there are exactly n cubefull numbers between k^3 and (k+1)^3.

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