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%I #10 Sep 28 2022 03:53:10
%S 1,0,1,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,
%T 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,1,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0
%N a(n) = A071626(n+1) - A071626(n).
%C Conjectures:
%C 1) The only distinct terms of this sequence are -1, 0 and 1.
%C 2) The asymptotic density of the occurrences of 0 is 0.943..., and the asymptotic densities of the occurrences of -1 and 1 are equal and the common value is 0.028... .
%C These conjectures are empirically supported by the following numbers of occurrences of the terms -1, 0 and 1 up to n = 10^k:
%C k -1 0 1
%C - ------- ------- ------
%C 1 0 6 4
%C 2 1 87 12
%C 3 28 913 59
%C 4 362 9189 449
%C 5 3159 93430 3411
%C 6 27853 943571 28576
%C 7 281802 9434309 283889
%H Amiram Eldar, <a href="/A357385/b357385.txt">Table of n, a(n) for n = 1..10000</a>
%F a(p) = 0 for an odd prime p.
%t s[n_] := Length[Union[FactorInteger[n!][[;; , 2]]]]; s[1] = 0; Differences[Table[s[n], {n, 1, 100}]]
%Y Cf. A071626.
%K sign
%O 1
%A _Amiram Eldar_, Sep 26 2022