A262084 -id:A262084

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A056772

Numbers n such that phi(n+4) = phi(n) + 4.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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)

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..1000

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

Cf. A015913 (sigma(n+4) = sigma(n) + 4).

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

KEYWORD

nonn

AUTHOR

Labos Elemer, Aug 17 2000

STATUS

approved

A262085

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

Wikipedia, Euler's totient function

EXAMPLE

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).

MAPLE

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

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

Cf. A000010.

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

KEYWORD

nonn,easy

AUTHOR

Kevin J. Gomez, Sep 10 2015

STATUS

approved

A262086

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Wikipedia, Euler's totient function

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A000010, A023203.

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

KEYWORD

nonn,easy

AUTHOR

Kevin J. Gomez, Sep 10 2015

STATUS

approved

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