A240591 The smaller of a pair of successive powerful numbers (A001694) without any prime number between them.

8, 25, 32, 121, 288, 675, 1331, 1369, 1936, 2187, 2700, 3125, 5324, 6724, 9800, 10800, 12167, 15125, 32761, 39200, 48668, 70225, 79507, 88200, 97336, 107648, 143641, 156800, 212521, 228484, 235224, 280900, 312481, 332928, 456968, 465124, 574564, 674028, 744769, 829921, 830297, 857476, 877952, 940896

Offset: 1

Author

Antonio Roldán, Apr 08 2014

Keywords

• nonn

Examples

			25 is in the sequence because A001694(6)=25, A001694(7)=27, without primes between them.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..373 (terms below 10^15; terms 1..103 from Amiram Eldar, terms 104..235 from Chai Wah Wu)

Crossrefs

Supersequence of A060355.

Cf. A001694, A240590.

Programs

Mathematica

Select[Partition[Join[{1},Select[Range[10^6],Min@FactorInteger[#][[All, 2]]> 1&]],2,1],PrimePi[#[[1]]]==PrimePi[#[[2]]]&][[All,1]] (* Harvey P. Dale, Mar 28 2018 *)

PARI

ispowerful(n)={local(h);if(n==1,h=1,h=(vecmin(factor(n)[, 2])>1));return(h)}
nextpowerful(n)={local(k);k=n+1;while(!ispowerful(k),k+=1);return(k)}
{for(i=1,10^6,if(ispowerful(i),if(nextprime(i)>=nextpowerful(i),print1(i, ", "))))}

Python

from itertools import count, islice
from math import isqrt
from sympy import mobius, integer_nthroot, nextprime
def A240591_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 = 1
    for n in count(2):
        k = bisection(lambda x:f(x)+n,m,m)
        if nextprime(m) > k:
            yield m
        m = k
A240591_list = list(islice(A240591_gen(),30)) # Chai Wah Wu, Sep 14 2024