A322871 -id:A322871 - cp's OEIS frontend

A322873 Ordinal transform of A300721, which is Möbius transform of A060681.

1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 1, 4, 3, 4, 1, 4, 1, 5, 2, 5, 1, 6, 2, 6, 2, 3, 1, 7, 1, 1, 2, 7, 2, 4, 1, 8, 3, 2, 1, 5, 1, 3, 3, 9, 1, 3, 2, 8, 4, 4, 1, 6, 2, 5, 3, 10, 1, 4, 1, 11, 1, 5, 3, 4, 1, 6, 2, 7, 1, 6, 1, 12, 2, 4, 1, 7, 1, 7, 4, 13, 1, 8, 1, 14, 2, 1, 1, 5, 2, 3, 3, 15, 1, 8, 1, 8, 2, 2, 1, 9, 1, 3, 4

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

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

Cf. A060681, A300721, A081373, A303756, A322871, A322874.

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
A300721[n_] := Sum[MoebiusMu[n/d] A060681[d], {d, Divisors[n]}];
b[_] = 1;
a[n_] := a[n] = With[{t = A300721[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]))));
A300721(n) = sumdiv(n, d, moebius(n/d)*A060681(d));
v322873 = ordinal_transform(vector(up_to,n,A300721(n)));
A322873(n) = v322873[n];

A322870 Ordinal transform of A302043.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 3, 1, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 4, 3, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 4, 2, 1, 2, 1, 1, 1, 5, 2, 1, 1, 2, 2, 1, 1, 1, 3, 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. A078898, A302042, A302043, A322871.

up_to = 1024;
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A302043(n) = (n - A302042(n));
v322870 = ordinal_transform(vector(up_to,n,A302043(n)));
A322870(n) = v322870[n];

A322872 Ordinal transform of A171462, where A171462(n) = n - A052126(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 2, 3, 3, 1, 4, 1, 2, 2, 1, 1, 3, 2, 1, 3, 2, 1, 3, 1, 4, 2, 1, 3, 4, 1, 2, 3, 2, 1, 4, 1, 2, 5, 1, 1, 3, 2, 3, 1, 2, 1, 6, 1, 3, 1, 1, 1, 4, 1, 2, 2, 4, 3, 4, 1, 1, 2, 5, 1, 5, 1, 2, 6, 3, 2, 4, 1, 2, 3, 1, 1, 5, 2, 1, 2, 3, 1, 6, 3, 2, 1, 1, 2, 3, 1, 4, 3, 4, 1, 2, 1, 3, 4

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

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

Cf. A052126, A171462, A081373, A303756, A322871.

A171462[n_] := If[n == 1, 0, Module[{f = FactorInteger[n], p},
     p = f[[-1, 1]]; n(p-1)/p]];
b[_] = 0;
a[n_] := a[n] = With[{t = A171462[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 21 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; };
A171462(n) = if(1==n,0,(n-(n/vecmax(factor(n)[, 1]))));
v322872 = ordinal_transform(vector(up_to,n,A171462(n)));
A322872(n) = v322872[n];

cp's OEIS Frontend

A322873 Ordinal transform of A300721, which is Möbius transform of A060681.

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A322870 Ordinal transform of A302043.

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A322872 Ordinal transform of A171462, where A171462(n) = n - A052126(n).

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