A330747 -id:A330747 - cp's OEIS frontend

A049559 a(n) = gcd(n - 1, phi(n)).

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 22, 1, 4, 1, 2, 3, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 6, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4

Offset: 1

Labos Elemer, Dec 28 2000

nonn

For prime n, a(n) = n - 1. Question: do nonprimes exist with this property?
Answer: No. If n is composite then a(n) < n - 1. - Charles R Greathouse IV, Dec 09 2013
Lehmer's totient problem (1932): are there composite numbers n such that a(n) = phi(n)? - Thomas Ordowski, Nov 08 2015
a(n) = 1 for n in A209211. - Robert Israel, Nov 09 2015

			a(9) = 2 because phi(9) = 6 and gcd(8, 6) = 2.
a(10) = 1 because phi(10) = 4 and gcd(9, 4) = 1.

Richard K. Guy, Unsolved Problems in Number Theory, B37.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lehmer's Totient Problem

Cf. A000010, A002322, A039766, A063994, A160595, A209211, A219428, A264012, A264024, A280262, A283656, A283872, A284089, A284440, A318827, A318829, A318830, A330747 (ordinal transform), A340195.

Cf. also A009195, A058515, A058663, A187730, A258409, A339964, A340071, A340081, A340087 for more or less analogous sequences.

[Gcd(n-1, EulerPhi(n)): n in [1..80]]; // Vincenzo Librandi, Oct 13 2018

seq(igcd(n-1, numtheory:-phi(n)), n=1..100); # Robert Israel, Nov 09 2015

Table[GCD[n - 1, EulerPhi[n]], {n, 93}] (* Michael De Vlieger, Nov 09 2015 *)

a(n)=gcd(eulerphi(n),n-1) \\ Charles R Greathouse IV, Dec 09 2013

from sympy import totient, gcd
print([gcd(totient(n), n - 1) for n in range(1, 101)]) # Indranil Ghosh, Mar 27 2017

a(p^m) = a(p) = p - 1 for prime p and m > 0. - Thomas Ordowski, Dec 10 2013
From Antti Karttunen, Sep 09 2018: (Start)
a(n) = A000010(n) / A160595(n) = A063994(n) / A318829(n).
a(n) = n - A318827(n) = A000010(n) - A318830(n).
(End)
a(n) = gcd(A000010(n), A219428(n)) = gcd(A000010(n), A318830(n)). - Antti Karttunen, Jan 05 2021

A330756 Number of values of k, 1 <= k <= n, with A063994(k) = A063994(n), where A063994(n) = Product_{primes p dividing n} gcd(p-1, n-1).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 7, 1, 8, 2, 9, 1, 10, 1, 11, 3, 12, 1, 13, 4, 14, 3, 1, 1, 15, 1, 16, 5, 17, 6, 18, 1, 19, 7, 20, 1, 21, 1, 22, 1, 23, 1, 24, 2, 25, 8, 2, 1, 26, 9, 27, 10, 28, 1, 29, 1, 30, 11, 31, 2, 1, 1, 32, 12, 3, 1, 33, 1, 34, 13, 4, 14, 35, 1, 36, 4, 37, 1, 38, 3, 39, 15, 40, 1, 41, 2, 42, 16

Offset: 1

Antti Karttunen, Dec 30 2019

nonn

Ordinal transform of A063994.

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

Cf. A063994, A209211.

Cf. also A081373, A303756, A330747.

A063994[n_] := If[n==1, 1, Times @@ GCD[n-1, First /@ FactorInteger[n]-1]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A063994[n]}, b[t] = b[t]+1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

up_to = 65537;
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
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; };
v330756 = ordinal_transform(vector(up_to, n, A063994(n)));
A330756(n) = v330756[n];

A344774 Number of divisors d of n for which A049559(d) = A049559(n), where A049559(n) = gcd(n-1, phi(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 31 2021

nonn

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

Cf. A000010, A049559, A330747.

Cf. also A344773.

A049559(n) = gcd(n-1, eulerphi(n));
A344774(n) = { my(x=A049559(n)); sumdiv(n,d,A049559(d)==x); };

a(n) = Sum_{d|n} [A049559(d) = A049559(n)], where [ ] is the Iverson bracket.

a(n) <= A330747(n).

cp's OEIS Frontend

A049559 a(n) = gcd(n - 1, phi(n)).

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A330756 Number of values of k, 1 <= k <= n, with A063994(k) = A063994(n), where A063994(n) = Product_{primes p dividing n} gcd(p-1, n-1).

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A344774 Number of divisors d of n for which A049559(d) = A049559(n), where A049559(n) = gcd(n-1, phi(n)).

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