A345938 -id:A345938 - cp's OEIS frontend

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

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.

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.

A345948 a(n) = A344875(n) / gcd(A153151(n), A344875(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 4, 1, 14, 1, 18, 4, 1, 1, 24, 1, 28, 3, 10, 1, 30, 1, 36, 1, 14, 1, 24, 1, 1, 5, 16, 12, 8, 1, 54, 12, 20, 1, 36, 1, 70, 8, 22, 1, 62, 1, 72, 16, 28, 1, 78, 20, 18, 9, 28, 1, 56, 1, 90, 24, 1, 3, 12, 1, 112, 11, 24, 1, 120, 1, 108, 24, 42, 15, 72, 1, 124, 1, 40, 1, 84, 16, 126, 28, 50, 1, 96, 4, 22

Offset: 1

Antti Karttunen, Jul 01 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. A153151, A344875, A345947, A345949.

Cf. also A345938.

{1}~Join~Array[#2/GCD @@ {##} & @@ {Which[# < 2, #, IntegerQ[Log2@ #], 2 # - 1, True, # - 1], Times @@ Map[If[#1 == 2, 2^(#2 + 1) - 1, #1^#2 - 1] & @@ # &, FactorInteger[#]]} &, 91, 2] (* Michael De Vlieger, Jul 06 2021 *)

A153151(n) = if(!n,n,if(!bitand(n,n-1),(n+n-1),(n-1)));
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A345948(n) = { my(u=A344875(n)); (u/gcd(A153151(n), u)); };

a(n) = A344875(n) / A345947(n) = A344875(n) / gcd(A153151(n), A344875(n)).

a(2n-1) = A345938(2n-1), for all n >= 1.

cp's OEIS Frontend

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

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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

Keywords

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Crossrefs

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Mathematica

PARI

Formula

A345948 a(n) = A344875(n) / gcd(A153151(n), A344875(n)).

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Mathematica

PARI

Formula