A303758 -id:A303758 - cp's OEIS frontend

A303756 Number of values of k, 1 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

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

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of A002322.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002322.

Cf. also A081373, A303755, A303758.

a[n_] := With[{c = CarmichaelLambda[n]}, Select[Range[n], c == CarmichaelLambda[#]&] // Length];
Array[a, 1000] (* Jean-François Alcover, Sep 19 2020 *)

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; };
A002322(n) = lcm(znstar(n)[2]); \\ From A002322
v303756 = ordinal_transform(vector(up_to,n,A002322(n)));
A303756(n) = v303756[n];

Except for a(2) = 2, a(n) = A303758(n).

A303757 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A000010(k) = A000010(n), where A000010 is Euler totient function phi.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 1, 2, 1, 4, 1, 3, 2, 2, 1, 4, 1, 3, 2, 4, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 2, 4, 1, 4, 1, 4, 2, 5, 2, 2, 1, 6, 1, 2, 3, 2, 1, 5, 1, 3, 1, 6, 1, 7, 1, 4, 3, 5, 2, 8, 1, 4, 1, 4, 1, 9, 1, 3, 1, 5, 1, 10, 2, 2, 3, 2, 3, 5, 1, 4, 4, 6, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Apr 30 2018

nonn

Ordinal transform of f, where f(1) = 0 and f(n) = A000010(n) for n > 1.
After a(1)=1 and a(4)=2, the positions of the rest of records is given by A081375(n) = 6, 12, 30, 42, 72, 78, 84, 90, 190, ..., for n >= 3.
Apart from a(2) = 1, the other positions of 1's is given by A210719.

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

Cf. A000010, A081373, A081375, A210719, A303754, A303758, A303777.

With[{s = EulerPhi@ Range@ 105}, MapAt[# + 1 &, Table[Count[s[[2 ;; n]], ?(# == s[[n]] &)], {n, Length@ s}], 1]] (* _Michael De Vlieger, Nov 23 2018 *)

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; };
Aux303757(n) = if(1==n,0,eulerphi(n));
v303757 = ordinal_transform(vector(up_to,n,Aux303757(n)));
A303757(n) = v303757[n];

Except for a(2) = 1, a(n) = A081373(n).

cp's OEIS Frontend

A303756 Number of values of k, 1 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

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