A358173 First differences of A286708.

36, 28, 8, 36, 52, 4, 16, 9, 63, 36, 68, 8, 32, 9, 43, 16, 76, 72, 27, 1, 108, 16, 64, 36, 68, 4, 28, 89, 36, 27, 4, 69, 71, 27, 29, 20, 72, 77, 47, 32, 128, 36, 36, 136, 8, 56, 25, 91, 188, 8, 188, 92, 9, 99, 4, 40, 144, 28, 109, 62, 49, 64, 49, 18, 97, 11, 81

Offset: 1

Author

Michael De Vlieger, Nov 01 2022

Keywords

• nonn

Comments

Consider the sequence of powerful numbers A001694, superset of A246547, the sequence of composite prime powers. Let s = A001694(k) such that omega(s) > 1 be followed by t = A001694(k+1) such that omega(t) = 1.

Since A286708 = A001694 \ A246547, prime powers t are missing in A286708. We consider s = A286708(j) and note that the difference A286708(j+1) - A286708(j) > A001694(k+1) - A001694(k).

Therefore we see a subset S containing s in A286708 that plots "out of place" with respect to the complementary subset R = A286708 \ S; some of this subset S exceeds the maxima of R in the scatterplot of this sequence. The plot of the R resembles the scatterplot of A001694.

Examples

			The number 36 is the smallest powerful number that is not a prime power; the next powerful number that is not a prime power is 72, and their difference is 36, hence a(1) = 36.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Scatterplot of a(n), n = 1..2^16, highlighting in red terms s that in A001694 are succeeded by a prime power t.

Crossrefs

Cf. A001694, A053707, A076446, A246547, A286708.

Programs

Mathematica

With[{nn = 2^25}, Differences@ Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], ! PrimePowerQ[#] &]]

Python

from math import isqrt
from sympy import integer_nthroot, primepi, mobius
def A358173(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+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length())), 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