A202855 -id:A202855 - cp's OEIS frontend

A203966 Numbers n such that phi(n) divides n+1, where phi is Euler's totient function (A000010).

1, 2, 3, 15, 255, 65535, 83623935, 4294967295, 6992962672132095

Offset: 1

José María Grau Ribas, Jan 08 2012

Numbers k such that A060473(k) = 1. - Robert G. Wilson v, Jul 06 2014
Except for a(2), all terms are odd. - Chai Wah Wu, Jun 06 2017
Since gcd(phi(n),n) = 1, all terms are squarefree. Then, for n = p1 * ... * pk with primes p1 < ... < pk, (n+1)/phi(n) is very close to p1/(p1-1)*...*pk/(p1-1). Since p/(p-1) is decreasing as p grows, having (n+1)/phi(n) = 3 implies that p1 >= 5 and further that n >= 2.4*10^56 is a product of at least 33 primes. Similarly, having (n+1)/phi(n) >= 4 implies that n >= 1.6*10^30 is a product of at least 21 primes. Hence, the terms of this sequence below 1.6*10^30 have (n+1)/phi(n) = 2 and thus coincide with A050474. - Max Alekseyev, Jan 30 2022

			15 is in the sequence because phi(15) = 8, and 8 divides 16 = 15 + 1 evenly.

D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.

Cf. A060473, A000010, A202855.

Superset of A050474.

Select[Range[100000], Divisible[#+1, EulerPhi[#]]&]

a(8) from Donovan Johnson, Jan 13 2012

a(9) confirmed by Max Alekseyev, Jan 30 2022

A050474 Solutions to 2*phi(x) = x+1.

Original entry on oeis.org

1, 3, 15, 255, 65535, 83623935, 4294967295, 6992962672132095

Offset: 1

Jud McCranie, Dec 24 1999

If n is in the sequence and n+2 is prime then m=n*(n+2) is in the sequence because 2*phi(m) = 2*phi(n*(n+2)) = 2*phi(n)*(n+1) = (n+1)^2 = m+1. We can obtain the terms 3, 15, 255, 65535 & 4294967295 from 1 (the first term) in this way. Also since 83623935 is a term and 83623935+2 is prime 83623935*(83623935+2)=6992962672132095 is in the sequence. So 1 and 83623935 are the only known independent terms and next term of this sequence if it exists is the third such term. - Farideh Firoozbakht, May 01 2007
The next term, if it exists, has at least 7 distinct prime factors (see Beiler, p. 92). - Jud McCranie, Dec 13 2012
From Chris Boyd, Mar 22 2015: (Start)
Solutions to k*phi(x) = x + 1, including a(1) - a(8), were published in 1932 by D. H. Lehmer. In the paper's summing up, "3*5*353*929" (= 4919055) was printed in error; it should have read "3*5*17*353*929" (= 83623935), i.e., a(6). This error has been propagated in several subsequent texts, including Wong's thesis.
Lehmer identified solutions where x has fewer than 7 distinct prime factors. Wong showed that no additional solutions exist unless x has at least 8 distinct prime factors. It appears not to be excluded by either author that an unidentified solution < a(8) with 8 or more distinct prime factors may exist. (End)
There are no other terms below 10^25. - Max Alekseyev, May 04 2024

			2*phi(15) = 2*8 = 15 + 1, so 15 is a member of the sequence.

A. H. Beiler, Recreations in the Theory of Numbers, page 92.

D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
E. Wong, Computations on normal families of primes, Simon Fraser University, 1997, MSc thesis.

Subsequence of A203966.

Cf. A000010, A129613, A129614, A129615, A202855.

[n: n in [1..2*10^6] | 2*EulerPhi(n) eq (n+1)]; // Vincenzo Librandi, Mar 22 2015

Select[Range[700000], (# + 1)== 2 EulerPhi[#] &] (* Vincenzo Librandi, Mar 22 2015 *)

is_A050474(n)=if(2*eulerphi(n)==n+1,1,0) \\ Chris Boyd, Mar 22 2015

A number n is in the sequence iff phi(n^2)=1+2+3+...+n because n is in the sequence <=> 2*phi(n)=n+1 <=> n*phi(n)=n*(n+1)/2 <=> phi(n^2)=1+2+3++...+n. For n=1,2,...,5, a(n)=2^2^(n-1)-1. - Farideh Firoozbakht, Jan 26 2006

A207574 Numbers k such that phi(k)+2 divides k+2.

Original entry on oeis.org

1, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 390, 394, 398, 422, 446, 454, 458, 466, 478

Offset: 1

José María Grau Ribas, Feb 19 2012

Consists of the even semiprimes (other than 4) together with A207575. - Charles R Greathouse IV, Jul 15 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A207575, A202855, A203966.

Select[Range[1000],Divisible[#+2,EulerPhi[#]+2]&]

isA207574(n)={
        if( (n+2) % (eulerphi(n)+2) ==0,1,0) ;
}
{
for(n=1,200,
        if (isA207574(n) , print(n)) ;
) ;
} /* R. J. Mathar, Feb 23 2012 */

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

A226104 Numbers n such that phi(n)+3 divides n+3.

Original entry on oeis.org

1, 30, 42, 66, 78, 102, 114, 138, 174, 186, 195, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398

Offset: 1

José María Grau Ribas, May 26 2013

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010 (Euler's totient function).
Cf. A207574 (numbers n such that phi(n)+2 divides n+2).
Cf. A202855 (numbers n such that phi(n)-1 divides n).
Cf. A203966 (numbers n such that phi(n) divides n+1).

Select[Range[2000], IntegerQ[(# + 3)/(EulerPhi[#] + 3)] &]

is(n)=(n+3)%(eulerphi(n)+3)==0 \\ Charles R Greathouse IV, Nov 13 2013

from sympy import totient as phi
def ok(n): return (n+3) % (phi(n)+3) == 0
print([m for m in range(1, 1400) if ok(m)]) # Michael S. Branicky, Jan 30 2021

A207667 Numbers n such that phi(n) divides n+2.

Original entry on oeis.org

1, 2, 4, 6, 10, 30, 70, 510, 2590, 131070, 3359230, 167247870, 8589934590, 13985925344264190

Offset: 1

José María Grau Ribas, Feb 19 2012

Contains 2 * A203966 as a subsequence. - Max Alekseyev, Oct 27 2023

Cf. A000010, A050474, A207574, A202855, A203966.

Select[Range[50000000],Divisible[#+2,EulerPhi[#]]&]

a(12)-a(13) from Donovan Johnson, Mar 01 2012

a(14) from Max Alekseyev, Oct 27 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

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A203966 Numbers n such that phi(n) divides n+1, where phi is Euler's totient function (A000010).

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A050474 Solutions to 2*phi(x) = x+1.

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A207574 Numbers k such that phi(k)+2 divides k+2.

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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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A226104 Numbers n such that phi(n)+3 divides n+3.

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A207667 Numbers n such that phi(n) divides n+2.

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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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Mathematica

PARI

Extensions