A331177 -id:A331177 - cp's OEIS frontend

A331175 Number of values of k, 1 <= k <= n, with A109395(k) = A109395(n), where A109395(n) = n/gcd(n, phi(n)).

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A109395.

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

Cf. A000010, A003277, A009195, A109395.

Cf. also A081373, A330746, A331177.

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; };
A109395(n) = n/gcd(n, eulerphi(n));
v331175 = ordinal_transform(vector(up_to, n, A109395(n)));
A331175(n) = v331175[n];

For n >= 1, a(2^n) = n, a(A003277(n)) = 1.

A319677 Denominator of A047994(n)/n where A047994 is the unitary totient function.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 2, 13, 7, 15, 16, 17, 9, 19, 5, 7, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 32, 33, 17, 35, 3, 37, 19, 13, 10, 41, 7, 43, 22, 45, 23, 47, 8, 49, 25, 51, 13, 53, 27, 11, 4, 19, 29, 59, 5, 61, 31, 21, 64, 65, 33, 67, 17, 69, 35, 71

Offset: 1

Michel Marcus, Sep 26 2018

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

Cf. A047994, A030163, A305678, A319481, A319676 (numerators), A323409, A331177 (ordinal transform).

Cf. A002110, A060753.

uphi[n_] := Product[{p, e} = pe; p^e - 1, {pe, FactorInteger[n]}];
a[n_] := Denominator[uphi[n]/n];
Array[a, 100] (* Jean-François Alcover, Jan 10 2022 *)

a(n)=my(f=factor(n)~); denominator(prod(i=1, #f, f[1, i]^f[2, i]-1)/n);

a(p) = p, for p prime.
a(A002110(n)) = A060753(n).
a(n) = n / A323409(n) = n / gcd(n, A047994(n)). - Antti Karttunen, Jan 11 2020

A330739 Number of values of k, 1 <= k <= n, with A047994(k) = A047994(n), where A047994 is unitary totient function uphi(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A047994.

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

Cf. A047994.

Cf. also A081373 (ordinal transform of Euler totient function phi), A331177.

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[1, i]^f[2, i])-1); };
v330739 = ordinal_transform(vector(up_to, n, A047994(n)));
A330739(n) = v330739[n];

cp's OEIS Frontend

A331175 Number of values of k, 1 <= k <= n, with A109395(k) = A109395(n), where A109395(n) = n/gcd(n, phi(n)).

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A319677 Denominator of A047994(n)/n where A047994 is the unitary totient function.

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A330739 Number of values of k, 1 <= k <= n, with A047994(k) = A047994(n), where A047994 is unitary totient function uphi(n).

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