A076446 Differences of consecutive powerful numbers (definition 1).

3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 8, 9, 19, 8, 13, 4, 3, 16, 25, 27, 4, 16, 9, 18, 13, 32, 1, 35, 19, 18, 31, 8, 32, 9, 43, 16, 12, 17, 47, 49, 23, 27, 1, 53, 55, 16, 41, 23, 36, 61, 7, 4, 28, 24, 65, 36, 27, 4, 69, 71, 27, 8, 21, 17, 3, 72, 77, 47, 32, 81, 47, 36, 36, 49, 87, 8

Offset

1,1

Comments

The term 1 appears infinitely often. Erdos conjectured that two consecutive 1's do not occur. (see Guy).

References

R. K. Guy, Unsolved Problems in Number Theory, B16

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Powerful numbers

Formula

a(n) = A001694(n+1) - A001694(n).

Example

The first two powerful numbers are 1 and 4, there difference is 3, so a(1)=3.

Mathematica

Differences[Join[{1}, Select[Range[2000], Min[FactorInteger[#][[All, 2]]]>1&]]] (* Harvey P. Dale, Aug 27 2017 *)

Prog

(Haskell)
a076446 n = a076446_list !! (n-1)
a076446_list = zipWith (-) (tail a001694_list) a001694_list
-- Reinhard Zumkeller, Nov 30 2012
(Python)
from math import isqrt
from sympy import mobius, integer_nthroot
def A076446(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
    def f(x):
        c, l = n+x, 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
    return -(a:=bisection(f, n, n))+bisection(lambda x:f(x)+1, a, a) # Chai Wah Wu, Sep 10 2024

Crossrefs

Cf. A001694, A076444.

Sequence in context: A202500 A016607 A262216 * A053289 A076412 A053707

Adjacent sequences: A076443 A076444 A076445 * A076447 A076448 A076449

Keyword

nonn

Author

Jud McCranie, Oct 15 2002

Status

approved