A303756 -id:A303756 - cp's OEIS frontend

A330756 Number of values of k, 1 <= k <= n, with A063994(k) = A063994(n), where A063994(n) = Product_{primes p dividing n} gcd(p-1, n-1).

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

Offset: 1

Antti Karttunen, Dec 30 2019

nonn

Ordinal transform of A063994.

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

Cf. A063994, A209211.

Cf. also A081373, A303756, A330747.

A063994[n_] := If[n==1, 1, Times @@ GCD[n-1, First /@ FactorInteger[n]-1]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A063994[n]}, b[t] = b[t]+1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

up_to = 65537;
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
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; };
v330756 = ordinal_transform(vector(up_to, n, A063994(n)));
A330756(n) = v330756[n];

A344772 Ordinal transform of infinitary phi, A091732.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2021

Number of values of k, 1 <= k <= n, with A091732(k) = A091732(n).

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

Cf. A091732, A050376, A302777.

Cf. also A081373 (ordinal transform of phi), A303756 (of Carmichael's lambda), A330739 (of unitary phi).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1);
b[_] = 0;
a[n_] := a[n] = With[{t = iphi[n]}, b[t] = b[t] + 1];
Array[a, 105] (* Jean-François Alcover, Dec 27 2021, after Amiram Eldar in A091732 *)

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; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
v344772 = ordinal_transform(vector(up_to,n,A091732(n)));
A344772(n) = v344772[n];

cp's OEIS Frontend

A330756 Number of values of k, 1 <= k <= n, with A063994(k) = A063994(n), where A063994(n) = Product_{primes p dividing n} gcd(p-1, n-1).

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