A262085 -id:A262085 - cp's OEIS frontend

A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

3, 7, 12, 13, 18, 19, 24, 28, 36, 37, 40, 43, 66, 67, 79, 88, 97, 103, 109, 124, 127, 163, 184, 193, 223, 229, 232, 277, 307, 313, 328, 349, 379, 397, 424, 439, 457, 463, 487, 499, 508, 613, 643, 664, 673, 712, 739, 757, 769, 823, 853, 859, 877, 883, 904, 907

Offset: 1

Labos Elemer, Aug 17 2000

nonn

In contrast with A015913, composite solutions are not rare. Prime solutions are common.
From Kevin J. Gomez, Mar 02 2016: (Start)
Composite solutions have two known forms:
  n such that n = 4 * (2^p - 1) where 2^p - 1 is a Mersenne prime. (A001348)
  n such that n = 8q where q is a Sophie Germain prime. (A005394)
There are composite solutions (such as 36) that do not fit either of these forms. (End)

			For k = 1048: phi(1048) = 520, phi(1048+4) = 524.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Cf. A000010, A015913 (sigma(k+4) = sigma(k) + 4).

Cf. A001838 (k=2), this sequence (k=4), A262084 (k=6), A262085 (k=8), A262086 (k=10).

[n: n in [1..1000] | EulerPhi(n+4) eq EulerPhi(n)+4]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 4)== EulerPhi[#] + 4 &] (* Vincenzo Librandi, Sep 11 2015 *)
Position[Partition[EulerPhi[Range[1000]],5,1],?(#[[1]]+4==#[[5]]&),1, Heads-> False]//Flatten (* _Harvey P. Dale, Dec 18 2019 *)

isok(n) = eulerphi(n+4) == eulerphi(n) + 4; \\ Michel Marcus, Sep 11 2015

A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

Original entry on oeis.org

5, 7, 11, 13, 17, 21, 23, 31, 37, 40, 41, 47, 53, 56, 61, 67, 73, 83, 88, 97, 98, 101, 103, 107, 131, 136, 151, 152, 156, 157, 167, 173, 191, 193, 223, 227, 233, 237, 248, 251, 257, 263, 271, 277, 296, 307, 311, 328, 331, 347, 353, 367, 373, 376, 383, 433, 443

Offset: 1

Kevin J. Gomez, Sep 10 2015

The majority of solutions can be predicted by known properties of the equality. There are several solutions that do not fit these parameters.
An odd natural number m is a solution if m and m + 6 are both prime (sexy primes) (A023201).
Among the solutions for even natural numbers are all m = 8*p with odd primes p such that 4*p+3 is a prime number. Proof: From A000010 we can learn that the formula phi(p*2) = floor(((2 + p - 1) mod p)/(p - 1)) + p - 1 is known. If we define p = 4*q+3 and m = 8*q and insert, we will obtain phi(8*q+6) = 4*q+2. Also it is known that phi(8*q) = 4*q-4 if q is any odd prime. - Thomas Scheuerle, Dec 20 2024

			5 is a term since phi(5+6) = 10 = 6 + 4 = phi(5) + 6.

Peter Kagey, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010, A023201.

Cf. A001838 (k=2), A056772 (k=4), A262085 (k=8), A262086 (k=10).

[n: n in [1..500] | EulerPhi(n+6) eq EulerPhi(n)+6]; // Vincenzo Librandi, Sep 11 2015

Select[Range@500, EulerPhi@(# + 6)== EulerPhi[#] + 6 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 6) == eulerphi(n) + 6 \\ Anders Hellström, Sep 11 2015

[n for n in [1..1000] if euler_phi(n+6)==euler_phi(n)+6] # Tom Edgar, Sep 10 2015

A262086 Numbers k such that phi(k + 10) = phi(k) + 10, where phi(k) = A000010(k) is Euler's totient function.

Original entry on oeis.org

3, 7, 13, 19, 31, 36, 37, 43, 61, 73, 79, 97, 103, 127, 139, 157, 163, 181, 223, 229, 241, 271, 283, 307, 337, 349, 373, 379, 409, 421, 433, 439, 457, 499, 547, 577, 607, 631, 643, 673, 691, 709, 733, 751, 787, 811, 829, 853, 877, 919, 937, 967

Offset: 1

Kevin J. Gomez, Sep 10 2015

The only composite term less than 10^11 is 36. - Giovanni Resta, Sep 14 2015

			3 is in the sequence since phi(13) = phi(3) + 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)
Wikipedia, Euler's totient function.

Cf. A000010, A023203.

Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262085 (k=8), this sequence (k=10).

[n: n in [1..1000] | EulerPhi(n+10) eq EulerPhi(n)+10]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 10) == EulerPhi[#] + 10 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 10) == eulerphi(n) + 10 \\ Anders Hellström, Sep 11 2015

cp's OEIS Frontend

A056772 Numbers k such that phi(k+4) = phi(k) + 4, where phi(k) = A000010(k) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

A262086 Numbers k such that phi(k + 10) = phi(k) + 10, where phi(k) = A000010(k) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI