A350777 -id:A350777 - cp's OEIS frontend

A068494 a(n) = n mod phi(n).

0, 0, 1, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 7, 0, 1, 0, 1, 4, 9, 2, 1, 0, 5, 2, 9, 4, 1, 6, 1, 0, 13, 2, 11, 0, 1, 2, 15, 8, 1, 6, 1, 4, 21, 2, 1, 0, 7, 10, 19, 4, 1, 0, 15, 8, 21, 2, 1, 12, 1, 2, 27, 0, 17, 6, 1, 4, 25, 22, 1, 0, 1, 2, 35, 4, 17, 6, 1, 16, 27, 2, 1, 12, 21, 2, 31, 8, 1, 18, 19, 4

Offset: 1

Benoit Cloitre, Mar 11 2002

By Lehmer's Conjecture, when n > 2 then a(n) = 1 if and only if n is prime. The Notices article states "Lehmer's Conjecture (1932). phi(n) | (n-1) if and only if n is prime." - Michael Somos, Oct 14 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. H. Bailey and J. M. Borwein, Exploratory Experimentation and Computation, Notices of A. M. S. 58 (2011) 1410-1419, see p. 1416.

Positions of particular numbers: 0: A007694, 1 (conjectured): A065091, 3: A350777\{1, 2, 3}.

Cf. A055516.

a068494 n = mod n $ a000010 n  -- Reinhard Zumkeller, Oct 14 2011

[n mod EulerPhi(n): n in [1..100]]; // Vincenzo Librandi, Jul 19 2015

Table[Mod[n, EulerPhi[n]], {n, 100}] (* Alonso del Arte, Feb 15 2013 *)

PARI
```
for(n=1,150,print1(n%eulerphi(n),","))
```

{a(n) = n % eulerphi(n)}; /* Michael Somos, Oct 14 2011 */

b^(n - a(n)) == 1 (mod n) for every b coprime to n. - Thomas Ordowski, Jun 30 2017

A207575 Numbers k such that phi(k) + 2 divides k + 2 and k is not twice a prime.

Original entry on oeis.org

1, 390, 10374, 2283934267736070, 7316037865689066623729670

Offset: 1

José María Grau Ribas, Feb 19 2012

Contains 2 * terms t of A350777 such that (t-3)/phi(t) = 2. - Max Alekseyev, Oct 26 2023

Cf. A000010, A207574, A202855, A203966, A350777.

Select[Range[20000000], !PrimeQ[#/2] && Divisible[#+2, EulerPhi[#]+2]&]

for(n=1,1e5,if((n+2)%(eulerphi(n)+2)==0&&(n%2||!isprime(n/2)), print1(n", "))) \\ Charles R Greathouse IV, Mar 02 2012

a(4)-a(5) from Max Alekseyev, Nov 06 2023

A226105 Numbers k such that phi(k)+3 divides k+3, excluding numbers of the form 6*p for a prime p.

Original entry on oeis.org

1, 195, 5187, 1141967133868035, 3658018932844533311864835

Offset: 1

José María Grau Ribas, May 26 2013

Terms having (k+3)/(phi(k)+3) = 2 are shared with A350777. - Max Alekseyev, Oct 26 2023

Set difference of A226104 and 6 * A000040.

Cf. A202855, A203966, A207575, A207667, A350777.

Select[Range[10000000], !PrimeQ[#/6] && IntegerQ[(# + 3)/(EulerPhi[#] + 3)] &]

for(n=1,10^8, if( (n+3)%(eulerphi(n)+3)==0 && (n%6 || !isprime(n\6)), print(n)));

Edited and a(4)-a(5) added by Max Alekseyev, Nov 05 2023

A039777 Integers m such that phi(m) is equal to the sum of (the product of prime factors) and (the product of exponents) of m-1.

Original entry on oeis.org

2, 5, 21, 45, 285, 765, 27645, 196605, 41067645, 72787965, 250871805, 4295098365, 12884901885, 23307153405, 172130669565, 1766029428523005, 20978888016396285

Offset: 1

Olivier Gérard

No other terms below 10^24. Some large terms: 1039619980803100740810795122685, 32576974833437288924302842789885. - Max Alekseyev, Jul 28 2024
All listed terms represent solutions to phi(m) = (m+3)/2 such that (m-1)/2 is an even squarefree number. Cf. A350777. - Max Alekseyev, Jul 21 2024
a(1)=2 is the only even term below 10^100000. - Max Alekseyev, Jul 22 2024

			21 is a term since 21-1 = 2^2*5^1 and (2*5)+(2*1) = 12 = phi(21).

Cf. A000010, A039696, A350777.

More terms from Jud McCranie
Corrected example and a(11)-a(14) from Donovan Johnson, Nov 14 2010
a(15)-a(17) from Max Alekseyev, Jul 21 2024

cp's OEIS Frontend

A068494 a(n) = n mod phi(n).

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A207575 Numbers k such that phi(k) + 2 divides k + 2 and k is not twice a prime.

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A226105 Numbers k such that phi(k)+3 divides k+3, excluding numbers of the form 6*p for a prime p.

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A039777 Integers m such that phi(m) is equal to the sum of (the product of prime factors) and (the product of exponents) of m-1.

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