A322873 -id:A322873 - cp's OEIS frontend

A322871 Ordinal transform of A060681, where A060681(n) = n - A032742(n).

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

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A032742, A060681, A081373, A303756, A322870, A322872, A322873, A322874.

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
b[_] = 1;
a[n_] := a[n] = With[{t = A060681[n]}, b[t]++];
a /@ Range[1, 105] (* Jean-François Alcover, Dec 19 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A060681(n) = if(1==n,0,(n-(n/vecmin(factor(n)[, 1]))));
v322871 = ordinal_transform(vector(up_to,n,A060681(n)));
A322871(n) = v322871[n];

A300721 Möbius transform of A060681, the largest difference between consecutive divisors of n (ordered by size).

Original entry on oeis.org

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, 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, 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

Offset: 1

Antti Karttunen, Mar 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

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

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
a[n_] := Sum[MoebiusMu[n/d]*A060681[d], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Jan 07 2022 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A300721(n) = sumdiv(n,d,moebius(n/d)*A060681(d));

a(n) = Sum_{d|n} A008683(n/d)*A060681(d).
a(n) = A060681(n) - A300722(n).
a(n) = A000010(n) - A300236(n).

A322874 Ordinal transform of A007431, which is Möbius transform of Euler phi.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 1, 1, 1, 3, 1, 4, 1, 4, 2, 2, 1, 5, 1, 3, 2, 6, 1, 2, 1, 7, 1, 3, 1, 8, 1, 1, 2, 9, 2, 3, 1, 10, 2, 1, 1, 11, 1, 3, 2, 12, 1, 4, 1, 13, 3, 3, 1, 14, 2, 1, 2, 15, 1, 4, 1, 16, 1, 2, 1, 17, 1, 4, 2, 18, 1, 2, 1, 19, 3, 3, 2, 20, 1, 3, 2, 21, 1, 4, 3, 22, 3, 1, 1, 23, 1, 3, 2, 24, 2, 3, 1, 25, 3, 4, 1, 26, 1, 1, 5

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007431, A081373, A303756, A322873.

A007431[n_] := Sum[EulerPhi[d] MoebiusMu[n/d], {d, Divisors[n]}];
b[_] = 0;
a[n_] := a[n] = With[{t = A007431[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A007431(n) = sumdiv(n,d,moebius(n/d)*eulerphi(d));
v322874 = ordinal_transform(vector(up_to,n,A007431(n)));
A322874(n) = v322874[n];

cp's OEIS Frontend

A322871 Ordinal transform of A060681, where A060681(n) = n - A032742(n).

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A300721 Möbius transform of A060681, the largest difference between consecutive divisors of n (ordered by size).

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A322874 Ordinal transform of A007431, which is Möbius transform of Euler phi.

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