A331175 -id:A331175 - cp's OEIS frontend

A331177 Number of values of k, 1 <= k <= n, with A319677(k) = A319677(n), where A319677(n) = n/gcd(n, uphi(n)), and uphi is unitary totient function (A047994).

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A319677.

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

Cf. A047994, A319677.

Cf. also A330739, A331175.

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; };
A047994(n) = { my(f=factor(n)); prod(i=1, #f~, (f[i, 1]^f[i, 2])-1); };
A319677(n) = n/gcd(n, A047994(n));
v331177 = ordinal_transform(vector(up_to, n, A319677(n)));
A331177(n) = v331177[n];

A330746 Number of values of k, 1 <= k <= n, with A017666(k) = A017666(n), where A017666(n) = n/gcd(n, sigma(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A017666.

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

Cf. A009194, A014567, A017666.
A left inverse of following sequences: A007691, A159907, A245775.
Cf. also A331175.

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; };
A017666(n) = (n/gcd(n, sigma(n)));
v330746 = ordinal_transform(vector(up_to, n, A017666(n)));
A330746(n) = v330746[n];

For all n >= 1, a(A014567(n)) = 1.

For all n >= 1, a(A007691(n)) = a(A159907(n)) = a(A245775(n)) = n.

A331178 Number of values of k, 1 <= k <= n, with A023900(k) = A023900(n), where A023900 is Dirichlet inverse of Euler totient function phi.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A023900.

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

Cf. A000010, A023900.

Cf. also A081373, A331175, A331179, A331180.

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; };
A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
v331178 = ordinal_transform(vector(up_to, n, A023900(n)));
A331178(n) = v331178[n];

A331182 Number of values of k, 1 <= k <= n, with A083254(k) = A083254(n), where A083254(n) = 2*phi(n) - n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A083254.

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

Cf. A000010, A083254.

Cf. also A081373, A331175, A331181.

Module[{b}, b[_] = 0;
a[n_] := With[{t = 2 EulerPhi[n] - n}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

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; };
A083254(n) = (2*eulerphi(n)-n);
v331182 = ordinal_transform(vector(up_to,n,A083254(n)));
A331182(n) = v331182[n];

A331179 Number of values of k, 1 <= k <= n, with A173557(k) = A173557(n), where A173557(n) = Product_{p-1 | p is prime and divisor of n}.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A173557.

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

Cf. A173557.

Cf. also A081373, A331175, A331178.

A173557[n_] := If[n == 1, 1, Times @@ (FactorInteger[n][[All, 1]] - 1)];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A173557[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

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; };
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
v331179 = ordinal_transform(vector(up_to, n, A173557(n)));
A331179(n) = v331179[n];

cp's OEIS Frontend

A331177 Number of values of k, 1 <= k <= n, with A319677(k) = A319677(n), where A319677(n) = n/gcd(n, uphi(n)), and uphi is unitary totient function (A047994).

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A330746 Number of values of k, 1 <= k <= n, with A017666(k) = A017666(n), where A017666(n) = n/gcd(n, sigma(n)).

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A331178 Number of values of k, 1 <= k <= n, with A023900(k) = A023900(n), where A023900 is Dirichlet inverse of Euler totient function phi.

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A331182 Number of values of k, 1 <= k <= n, with A083254(k) = A083254(n), where A083254(n) = 2*phi(n) - n.

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A331179 Number of values of k, 1 <= k <= n, with A173557(k) = A173557(n), where A173557(n) = Product_{p-1 | p is prime and divisor of n}.

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