A323409 -id:A323409 - cp's OEIS frontend

A323410 Unitary analog of cototient function A051953: a(n) = n - A047994(n).

0, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 6, 1, 8, 7, 1, 1, 10, 1, 8, 9, 12, 1, 10, 1, 14, 1, 10, 1, 22, 1, 1, 13, 18, 11, 12, 1, 20, 15, 12, 1, 30, 1, 14, 13, 24, 1, 18, 1, 26, 19, 16, 1, 28, 15, 14, 21, 30, 1, 36, 1, 32, 15, 1, 17, 46, 1, 20, 25, 46, 1, 16, 1, 38, 27, 22, 17, 54, 1, 20, 1, 42, 1, 48, 21, 44, 31, 18, 1, 58, 19, 26, 33, 48

Offset: 1

Antti Karttunen, Jan 15 2019

Antti Karttunen, Table of n, a(n) for n = 1..21210
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A047994, A065463.

Cf. also A034460, A051953, A323409, A323413.

a[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); a[1] = 0; Array[a, 100] (* Amiram Eldar, Apr 08 2023 *)

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); };
A323410(n) = (n-A047994(n));

a(n) = n - A047994(n), where A047994 is unitary phi.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065463 = 0.2955577... . - Amiram Eldar, Dec 15 2023

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

A345937 a(n) = gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 7, 8, 1, 10, 1, 12, 1, 2, 15, 16, 1, 18, 1, 4, 1, 22, 1, 24, 1, 26, 9, 28, 1, 30, 31, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 48, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 63, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 80, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4, 1, 2, 1, 96

Offset: 1

Antti Karttunen, Jun 29 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A047994, A345938, A345939.

Cf. also A049559, A340087, A323409, A345947.

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A345937(n) = gcd(n-1, A047994(n));

a(n) = gcd(n-1, A047994(n)).
a(n) = A047994(n) / A345938(n).
a(n) = (n-1) / A345939(n), for n > 1.
a(2n-1) = A345947(2n-1), for n >= 1.

A344877 a(n) = gcd(n, A344875(n)), where A344875 is multiplicative with a(2^e) = 2^(1+e) - 1, and a(p^e) = p^e -1 for odd primes p.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 6, 1, 4, 3, 2, 1, 6, 1, 2, 1, 14, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 20, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 6, 5, 2, 3, 2, 1, 4, 1, 2, 3, 1, 1, 6, 1, 4, 1, 2, 1, 24, 1, 2, 3, 2, 1, 6, 1, 4, 1, 2, 1, 84, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 1, 4, 1, 6, 1, 4, 3

Offset: 1

Antti Karttunen, Jun 03 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A344875.

Cf. also A323409.

f[2, e_] := 2^(e + 1) - 1; f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 03 2021 *)

A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344877(n) = gcd(n, A344875(n));

A323406 Greatest common divisor of Product (p_i^e_i)-1 and Product (p_i^e_i)+1, when n = Product (p_i^e_i): a(n) = gcd(A047994(n), A034448(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 6, 8, 1, 2, 2, 2, 6, 4, 2, 2, 2, 2, 6, 2, 2, 2, 8, 2, 1, 4, 2, 24, 2, 2, 6, 8, 2, 2, 12, 2, 30, 4, 2, 2, 2, 2, 6, 8, 2, 2, 2, 8, 6, 4, 2, 2, 24, 2, 6, 16, 1, 12, 4, 2, 6, 4, 24, 2, 2, 2, 6, 8, 2, 12, 24, 2, 6, 2, 2, 2, 4, 4, 6, 8, 2, 2, 4, 8, 6, 4, 2, 24, 2, 2, 6, 40, 2, 2, 8, 2, 42, 48

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A047994.

Cf. also A009223, A066086, A126865, A322362, A323166, A323409.

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); };
A034448(n) = { my(f=factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323406(n) = gcd(A034448(n), A047994(n));

a(n) = gcd(A034448(n), A047994(n)), where A034448 is unitary sigma, and A047994 is unitary phi.

cp's OEIS Frontend

A323410 Unitary analog of cototient function A051953: a(n) = n - A047994(n).

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

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A345937 a(n) = gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

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A344877 a(n) = gcd(n, A344875(n)), where A344875 is multiplicative with a(2^e) = 2^(1+e) - 1, and a(p^e) = p^e -1 for odd primes p.

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A323406 Greatest common divisor of Product (p_i^e_i)-1 and Product (p_i^e_i)+1, when n = Product (p_i^e_i): a(n) = gcd(A047994(n), A034448(n)).

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