A262084 -id:A262084 - 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

A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

Original entry on oeis.org

3, 5, 11, 23, 24, 29, 36, 42, 48, 50, 53, 56, 59, 71, 72, 80, 89, 101, 102, 125, 131, 132, 149, 173, 176, 191, 230, 233, 248, 263, 269, 359, 368, 389, 401, 431, 449, 464, 479, 491, 563, 569, 593, 599, 638, 653, 656, 683, 701, 719, 743, 761, 821, 848, 911, 929, 983

Offset: 1

Kevin J. Gomez, Sep 10 2015

Sequence includes numbers n such that n and n + 8 are both prime (A023202).
Sequence also includes numbers n equal to 8*(a Mersenne prime) (cf A000668).
Sequence also includes n such that n/16 and n/8 + 1 are both odd primes.
Contains more composites than sequences A262084 and A262086. This is most likely due to the fact that 8 is a power of 2, as in A001838.

			3 since phi(11) = phi(3) + 8 (3 and 11 are both prime).
24 is a solution since phi(32) = phi(24) + 8 (24 is 8 * 3; 3 is a Mersenne prime).

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

Cf. A000010.

Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262086 (k=10).

[n: n in [1..1000] | EulerPhi(n+8) eq EulerPhi(n)+8]; // Vincenzo Librandi, Sep 11 2015

select(t -> numtheory:-phi(t+8) = numtheory:-phi(t)+8, [$1..1000]); # Robert Israel, Mar 04 2016

Select[Range@1000, EulerPhi@(# + 8)== EulerPhi[#] + 8 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 8) == eulerphi(n) + 8 \\ Anders Hellström, Sep 11 2015

[n for n in (1..1000) if euler_phi(n+8) == euler_phi(n)+8] # Bruno Berselli, Mar 04 2016

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

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

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A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

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

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