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Möbius transform of A060681, the largest difference between consecutive divisors of n (ordered by size).
3

%I #17 Jan 07 2022 03:23:07

%S 0,1,2,1,4,0,6,2,4,0,10,2,12,0,4,4,16,2,18,4,6,0,22,4,16,0,12,6,28,4,

%T 30,8,10,0,18,6,36,0,12,8,40,6,42,10,16,0,46,8,36,4,16,12,52,6,30,12,

%U 18,0,58,8,60,0,24,16,36,10,66,16,22,6,70,12,72,0,24,18,50,12,78,16,36,0,82,12,48,0,28,20,88,8,60,22,30,0,54,16

%N Möbius transform of A060681, the largest difference between consecutive divisors of n (ordered by size).

%H Antti Karttunen, <a href="/A300721/b300721.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = Sum_{d|n} A008683(n/d)*A060681(d).

%F a(n) = A060681(n) - A300722(n).

%F a(n) = A000010(n) - A300236(n).

%t A060681[n_] := n - n/FactorInteger[n][[1, 1]];

%t a[n_] := Sum[MoebiusMu[n/d]*A060681[d], {d, Divisors[n]}];

%t Array[a, 100] (* _Jean-François Alcover_, Jan 07 2022 *)

%o (PARI)

%o A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));

%o A060681(n) = (n-A032742(n));

%o A300721(n) = sumdiv(n,d,moebius(n/d)*A060681(d));

%Y Cf. A000010, A008683, A060681, A300236, A300722, A322873 (ordinal transform).

%K nonn

%O 1,3

%A _Antti Karttunen_, Mar 11 2018