A340089 -id:A340089 - cp's OEIS frontend

A340087 a(n) = gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

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

Cf. A091732, A340088, A340089.

Cf. also A049559.

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); };
A340087(n) = gcd(n-1, A091732(n));

a(n) = gcd(n-1, A091732(n)).
a(n) = A091732(n) / A340088(n).
For n > 1, a(n) = (n-1) / A340089(n).

A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 6, 1, 6, 4, 1, 1, 8, 1, 12, 3, 10, 1, 6, 1, 12, 8, 2, 1, 8, 1, 15, 5, 16, 12, 24, 1, 18, 12, 4, 1, 12, 1, 30, 8, 22, 1, 30, 1, 24, 16, 12, 1, 16, 20, 18, 9, 28, 1, 24, 1, 30, 24, 5, 3, 4, 1, 48, 11, 8, 1, 24, 1, 36, 24, 18, 15, 24, 1, 60, 1, 40, 1, 36, 16, 42, 28, 10, 1, 32, 4, 66

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Conjecture: a(n) = 1 iff n = 1 or in A050376. This is an infinitary analog of Lehmer's totient conjecture from 1932.

For all i, j > 1: a(i) = a(j) => A302777(i) = A302777(j), if the above conjecture holds.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Wikipedia, Lehmer's totient problem

Cf. A050376, A084400, A091732, A302777, A340087, A340089.

Cf. also A160595.

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); };
A340088(n) = { my(x=A091732(n)); (x/gcd(n-1, x)); };

a(n) = A091732(n) / A340087(n) = A091732(n) / gcd(n-1, A091732(n)).

For all n >= 1, a(A084400(n)) = 1.

A345939 a(n) = (n-1) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 1, 9, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 5, 21, 1, 23, 1, 25, 1, 3, 1, 29, 1, 1, 8, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 1, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 1, 81, 1, 83, 21, 85, 43, 87, 1, 89, 5, 91

Offset: 1

Antti Karttunen, Jun 29 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A047994, A345937, A345938.

Cf. also A160596, A340089, A345949.

uphi[1]=1;uphi[n_]:=Times@@(#[[1]]^#[[2]]-1&/@FactorInteger[n]);
a[n_]:=(n-1)/GCD[n-1,uphi[n]];Array[a,100] (* Giorgos Kalogeropoulos, Jul 02 2021 *)

A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A345939(n) = ((n-1) / gcd(n-1, A047994(n)));

a(n) = (n-1) / A345937(n) = (n-1) / gcd(n-1, A047994(n)).

a(2n-1) = A345949(2n-1), for n > 1.

cp's OEIS Frontend

A340087 a(n) = gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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PARI

Formula

A345939 a(n) = (n-1) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

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Author

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PARI

Formula