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

1,1

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)

Example

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

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 *)

Prog

(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

Crossrefs

Supersequence of A060355.

Cf. A001694, A240590.

Sequence in context: A081504 A030796 A266927 * A116086 A270739 A239582

Adjacent sequences: A240588 A240589 A240590 * A240592 A240593 A240594

Keyword

nonn

Author

Antonio Roldán, Apr 08 2014

Status

approved