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

1,1

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

Links

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

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

Example

a(1) = 4 = 8 - 4.

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 *)

Prog

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

Crossrefs

Cf. A068781, A013929, A132345.

Cf. A059956, A065474, A229099.

Sequence in context: A084118 A046071 A376353 * A376312 A362331 A376264

Adjacent sequences: A078144 A078145 A078146 * A078148 A078149 A078150

Keyword

easy,nonn

Author

Labos Elemer, Nov 26 2002

Extensions

Offset fixed by Reinhard Zumkeller, Apr 22 2012

Status

approved