A371190 The smaller of a pair of successive powerful numbers without a nonsquarefree number between them.

1, 4, 8, 25, 32, 288, 675, 968, 1152, 1369, 2700, 9800, 12167, 39200, 48668, 70225, 235224, 332928, 465124, 1331712, 1825200, 5724500, 7300800, 11309768, 78960996, 189750625, 263672644, 384199200, 592192224, 912670088, 1536796800, 2368768896, 4931691075, 5425069447, 8957108164

Offset: 1

Author

Amiram Eldar, Mar 14 2024

Keywords

• nonn

Examples

			1 is a term since 1 and 4 are successive powerful numbers and the numbers between them, 2 and 3, are both squarefree.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..64
• Index entries for sequences related to powerful numbers.

Crossrefs

Cf. A001694, A005117, A013929, A240591.

Programs

Mathematica

seq[max_] := Module[{pows = Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]], s = {}}, Do[If[AllTrue[Range[pows[[k]] + 1, pows[[k + 1]] - 1], SquareFreeQ], AppendTo[s, pows[[k]]]], {k, 1, Length[pows] - 1}]; s]; seq[10^10]

PARI

lista(mx) = {my(s = List(), is); for(j = 1, sqrtnint(mx, 3), for(i = 1, sqrtint(mx\j^3), listput(s, i^2 * j^3))); s = Set(s); for(i = 1, #s - 1, is = 1; for(k = s[i]+1, s[i+1]-1, if(!issquarefree(k), is = 0; break)); if(is, print1(s[i], ", ")));}

Python

from math import isqrt
from sympy import mobius, integer_nthroot
def A371190_gen(): # generator of terms
    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 = 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
    m, w = 1, 1
    for n in count(2):
        k = bisection(lambda x:f(x)+n,m,m)
        if (a:=squarefreepi(k))-w==k-1-m:
            yield m
        m, w = k, a # Chai Wah Wu, Sep 15 2024