A309353 Let d(1), d(2), ..., d(q) be the q divisors of a number m. The sequence lists the numbers m such that {phi(d(1)), phi(d(2)), ..., phi(d(q))} is the set of the divisors of a number k, where phi is the Euler totient function.

1, 2, 3, 4, 6, 8, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 128, 160, 192, 240, 255, 256, 272, 320, 340, 384, 408, 480, 510, 512, 544, 640, 680, 768, 816, 960, 1020, 1024, 1088, 1280, 1360, 1536, 1632, 1920, 2040, 2048, 2176, 2560, 2720, 3072

Offset

1,2

Comments

The sequence is infinite because the powers of 2 are in the sequence.

Conjecture: the corresponding numbers k of the sequence is a sequence b(n) of powers of 2.

The sequence b(n) begins with 1, 1, 2, 2, 2, 4, 4, 8, 8, 8, 8, 8, 16, 16, 16, 16, 32, 32, 32, 32, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 512, 512, 1024, 1024, 1024, 1024, ...

The number of terms of the form 2^i is given by the sequence c(i,2^i) = 2, 3, 2, 5, 4, 4, 4, 9, 8, 8, 8, 8, 8, 8, 8, 17, ... for i = 0, 1, 2, ...

Property of the prime factors of a(n) for n > 1:

We observe periodic sequences of prime factors of the form 2^m + 1.

+---------------------------------------+------------------------------+

| subsequences of consecutive | corresponding prime factors |

| values of a(n) | |

+---------------------------------------+------------------------------+

|4, 6 |{2},{2, 3} |

|8, 12 |{2},{2, 3} |

|16, 20, 24, 30 |{2}, {2, 5}, {2, 3}, {2, 3, 5}|

|32, 40, 48, 60 |{2}, {2, 5}, {2, 3}, {2, 3, 5}|

|64, 80, 96, 120 |{2}, {2, 5}, {2, 3}, {2, 3, 5}|

|128, 160, 192, 240 |{2}, {2, 5}, {2, 3}, {2, 3, 5}|

|256, 272, 320, 340, 384, 408, 480, 510 |{2}, {2, 17},...,{2, 3, 5, 17}|

|512, 544, 640, 680, 768, 816, 960, 1020|{2}, {2, 17},...,{2, 3, 5, 17}|

|1024, 1088, 1280, 1360, 1536, ..., 2040|{2}, {2, 17},...,{2, 3, 5, 17}|

|2048, 2176, 2560, 2720, 3072, ..., 4080|{2}, {2, 17},...,{2, 3, 5, 17}|

|4096, 4352, 5120, 5440, 6144, ..., 8160|{2}, {2, 17},...,{2, 3, 5, 17}|

............................................................

Links

Amiram Eldar, Table of n, a(n) for n = 1..300

Example

272 is in the sequence because the divisors are {1, 2, 4, 8, 16, 17, 34, 68, 136, 272}, and {phi(1), phi(2), phi(4), phi(8), phi(16), phi(17), phi(34), phi(68), phi(136), phi(272)} = {1, 2, 4, 8, 16, 32, 64, 128} is the set of the divisors of 128.

Maple

with(numtheory):nn:=3000:
for n from 1 to nn do:
d:=divisors(n):n0:=nops(d):A:={op(d), d[n0]}:lst:={}:
  for k from 1 to n0 do:
   lst:=lst union {phi(d[k])}:
  od:
    n1:=nops(lst):B:={op(lst), lst[n1]}:
    d1:=divisors(lst[n1]):n2:=nops(d1):C:={op(d1), d1[n2]}:
     if B=C
      then
      printf(`%d, `, n):
      else
     fi:
od:

Mathematica

Select[Range[3100], (d = Union @ EulerPhi @ Divisors @ #) == Divisors[LCM @@ d] &] (* Amiram Eldar, Aug 02 2019 *)

Crossrefs

Cf. A000010, A000079 (powers of 2).

Sequence in context: A260653 A232711 A178751 * A081029 A300787 A171966

Adjacent sequences: A309350 A309351 A309352 * A309354 A309355 A309356

Keyword

nonn

Author

Michel Lagneau, Aug 02 2019

Status

approved