A078147 First differences of sequence of nonsquarefree numbers, A013929.

4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, 3, 1, 4, 1, 3, 4, 2, 2, 4, 2, 1, 1, 4, 4, 4, 4, 1, 3, 1, 3, 1, 1, 2, 4, 3, 1, 4, 4, 3, 1, 2, 2, 1, 3, 4, 2, 2, 4, 1, 2, 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, 1, 1, 1, 3, 2, 2, 4, 4, 1, 3, 4, 2, 2, 3

Offset: 1

Author

Labos Elemer, Nov 26 2002

Keywords

• easy
• nonn

Comments

Run lengths in A132345, apart from initial run of zeros. - Reinhard Zumkeller, Apr 22 2012

The asymptotic density of the occurrences of 1 in this sequence is density(A068781)/density(A013929) = (1 - 2 * A059956 + A065474)/A229099 = 0.272347... - Amiram Eldar, Mar 09 2021

Examples

			a(1) = 4 = 8 - 4.
		

Links

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

Crossrefs

Cf. A068781, A013929, A132345.

Cf. A059956, A065474, A229099.

Programs

Haskell

a078147 n = a078147_list !! (n-1)
a078147_list = zipWith (-) (tail a013929_list) a013929_list
-- Reinhard Zumkeller, Apr 22 2012

Mathematica

t=Flatten[Position[Table[MoebiusMu[w], {w, 1, 1000}], 0]]; t1=Delete[RotateLeft[t]-t, -1]
Differences[Select[Range[300],!SquareFreeQ[#]&]] (* Harvey P. Dale, May 07 2012 *)

PARI

lista(nn) = {my(prec=0); for (n=1, nn, if (!issquarefree(n), if (prec, print1(n-prec, ", ")); prec = n;););} \\ Michel Marcus, Mar 26 2020

Python

from math import isqrt
from sympy import mobius, factorint
def A078147(n):
    def f(x): return n+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)
    return next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values())) # Chai Wah Wu, Sep 10 2024

Formula

a(n) = A013929(n+1) - A013929(n).

a(n) = 1, 2, 3 or 4 since n = 4*k is always nonsquarefree.

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = Pi^2/(Pi^2-6) = 2.550546... - Amiram Eldar, Oct 21 2020

Extensions

Offset fixed by Reinhard Zumkeller, Apr 22 2012