A303753 Ordinal transform of cototient (A051953).

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

Offset: 1

Author

Antti Karttunen, Apr 30 2018

Keywords

• nonn

Comments

Number of values of k, 1 <= k <= n, with A051953(k) = A051953(n).

Links

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

Crossrefs

Cf. A051953, A065385 (gives a subset of the positions of ones).

Cf. also A081373, A303754.

Programs

Maple

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= numtheory[phi](n)-n; b(t):= b(t)+1
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 30 2018

Mathematica

b[_] = 0;
a[n_] := a[n] = With[{t = EulerPhi[n]-n}, b[t] = b[t]+1];
Array[a, 120] (* Jean-François Alcover, Dec 19 2021, after Alois P. Heinz *)

PARI

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)));
A303753(n) = v303753[n];

Formula

For all n >= 1, a(A000040(n)) = n.