A078147 - OEIS

%I #36 Sep 10 2024 14:22:19

%S 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,

%T 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,

%U 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

%N First differences of sequence of nonsquarefree numbers, A013929.

%C Run lengths in A132345, apart from initial run of zeros. - _Reinhard Zumkeller_, Apr 22 2012

%C 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

%H Reinhard Zumkeller, <a href="/A078147/b078147.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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

%e a(1) = 4 = 8 - 4.

%t t=Flatten[Position[Table[MoebiusMu[w], {w, 1, 1000}], 0]]; t1=Delete[RotateLeft[t]-t, -1]

%t Differences[Select[Range[300],!SquareFreeQ[#]&]] (* _Harvey P. Dale_, May 07 2012 *)

%o (Haskell)

%o a078147 n = a078147_list !! (n-1)

%o a078147_list = zipWith (-) (tail a013929_list) a013929_list

%o -- _Reinhard Zumkeller_, Apr 22 2012

%o (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

%o (Python)

%o from math import isqrt

%o from sympy import mobius, factorint

%o def A078147(n):

%o     def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o     m, k = n, f(n)

%o     while m != k: m, k = k, f(k)

%o     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

%Y Cf. A068781, A013929, A132345.

%Y Cf. A059956, A065474, A229099.

%K easy,nonn

%O 1,1

%A _Labos Elemer_, Nov 26 2002

%E Offset fixed by _Reinhard Zumkeller_, Apr 22 2012