A303753 -id:A303753 - cp's OEIS frontend

A303754 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A303753(k) = A303753(n), where A303753 is ordinal transform of cototient, A051953.

1, 1, 1, 2, 1, 3, 1, 2, 4, 5, 1, 6, 1, 3, 7, 2, 1, 8, 1, 4, 9, 3, 1, 10, 11, 12, 5, 6, 1, 13, 1, 4, 14, 15, 16, 17, 1, 18, 19, 7, 1, 20, 1, 5, 21, 2, 1, 22, 8, 9, 23, 24, 1, 25, 10, 11, 12, 6, 1, 26, 1, 7, 27, 3, 28, 29, 1, 13, 30, 14, 1, 31, 1, 32, 33, 34, 15, 35, 1, 16, 17, 36, 1, 37, 8, 18, 38, 9, 1, 39, 19, 4, 40, 2, 41, 42, 1, 43, 44, 20, 1, 45, 1, 21

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of f, where f(1) = 0 and f(n) = A303753(n) for n > 1.

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

Cf. A051953, A303753.

Cf. also A081373, A303757.

b[_] = 0;
A303753[n_] := A303753[n] = With[{t = EulerPhi[n] - n}, b[t] = b[t]+1];
f[n_] := If[n == 1, 0, A303753[n]];
Clear[b]; b[_] = 0;
a[n_] := a[n] = With[{t = f[n]}, b[t] = b[t]+1];
Array[a, 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; };
A051953(n) = (n - eulerphi(n));
v303753 = ordinal_transform(vector(up_to,n,A051953(n)));
Aux303754(n) = if(1==n,0,v303753[n]);
v303754 = ordinal_transform(vector(up_to,n,Aux303754(n)));
A303754(n) = v303754[n];

A322025 Ordinal transform of A322023.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 4, 1, 3, 1, 5, 1, 6, 2, 1, 2, 7, 1, 8, 1, 1, 4, 9, 1, 10, 2, 5, 2, 11, 1, 12, 1, 1, 3, 2, 1, 13, 3, 3, 1, 14, 1, 15, 1, 2, 6, 16, 1, 7, 1, 3, 3, 17, 2, 2, 1, 2, 8, 18, 1, 19, 9, 1, 4, 1, 1, 20, 3, 4, 2, 21, 1, 22, 2, 2, 2, 3, 1, 23, 1, 24, 3, 25, 1, 1, 4, 5, 3, 26, 1, 1, 5, 3, 10, 5, 1, 27, 3, 2, 1, 28, 1, 29, 1, 1

Offset: 1

Antti Karttunen, Dec 01 2018

nonn

Positions where 1, 2, 3, 4, 5, ... occur for the first time are 1, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, ... Note that this is not a subsequence of A000961; for example, 187 = 11*17 is a semiprime.

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

Cf. A000010, A002322, A081373, A303753, A303755, A303756, A303777, A322023.

\\ Needs also code from A322023.
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; };
v322025 = ordinal_transform(v322023);
A322025(n) = v322025[n];

cp's OEIS Frontend

A303754 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A303753(k) = A303753(n), where A303753 is ordinal transform of cototient, A051953.

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