A358089 First differences of A126706.

6, 2, 4, 4, 8, 4, 4, 1, 3, 2, 2, 2, 2, 4, 3, 5, 4, 3, 1, 4, 4, 4, 2, 2, 4, 2, 1, 1, 4, 4, 4, 4, 1, 3, 4, 2, 6, 3, 1, 4, 4, 3, 1, 2, 2, 1, 3, 4, 2, 2, 4, 3, 1, 3, 1, 4, 4, 4, 1, 3, 4, 2, 2, 4, 3, 1, 4, 4, 4, 4, 1, 3, 4, 2, 2, 4, 2, 2, 1, 3, 2, 2, 8, 1, 3, 4, 2

Offset: 1

Author

Michael De Vlieger, Oct 31 2022

Keywords

• nonn
• easy

Comments

A356322 relates to the first instances of exactly k consecutive 1's in this sequence.

a(n) - 1 = number of 0's between 1's in A355447.

For prime p, m such that m mod p^2, unless m = p^e, e > 1, is in A126706, as a consequence of definition of A126706. Therefore m <= 4 is common, m <= 9 much less so. Consequently, the arrangement of A126706 mod M for M in A061742 presents a quasi-modular pattern as seen in the example and raster link at A355447.

a(51265) = 7; m = 9 is not observed in the first 6577230 terms of the sequence, a dataset corresponding to terms k <= 2^24 in A126706.

Examples

			The smallest numbers that are neither squarefree nor a prime power are {12, 18, 20, 24, 28 ...}, therefore the first terms of this sequence are {6, 2, 4, 4, ...}.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A126706, A355447, A356322.

Programs

Mathematica

k = 0; Rest@ Reap[Do[If[And[#2 > 1, #1 != #2] & @@ {PrimeOmega[n], PrimeNu[n]}, Sow[n - k]; Set[k, n] ],{n, 270}] ][[-1, -1]]
(* Generate 317359 terms of this sequence from the image at A355447: *)
Differences@ Position[Flatten@ ImageData[Import["https://oeis.org/A355447/a355447_1.png", "PNG"]], 0.][[All, -1]]

Python

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A358089(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    r, k = n+1, f(n+1)+1
    while r != k:
        r, k = k, f(k)+1
    return r-m # Chai Wah Wu, Aug 15 2024