A306771 Numbers m such that m = i + j = i * k and phi(m) = phi(i) + phi(j) = phi(i) * phi(k) for some i, j, k, where phi is the Euler totient function A000010.

3, 15, 21, 33, 39, 51, 57, 69, 75, 87, 93, 105, 111, 123, 129, 141, 147, 159, 165, 177, 183, 195, 201, 213, 219, 231, 237, 249, 255, 267, 273, 285, 291, 303, 309, 321, 327, 339, 345, 357, 363, 375, 381, 393, 399, 411, 417, 429, 435, 447, 453, 465, 471, 483, 489

Offset

1,1

Comments

The 55 terms given in the data section are consistent with a definition "numbers congruent to 3 or 15 mod 18". - Peter Munn, May 12 2020

The observation above is true for the first 10^4 terms. - Amiram Eldar, Dec 08 2020

The observation above is true for every term; see link. - Flávio V. Fernandes, Apr 18 2022

A001748 \ {6, 9} is a subsequence because, for p prime >= 5, 3 * p = p + 2p = p * 3 and phi(3p) = phi(p) + phi(2p) = phi(p) * phi(3) = 2 * (p-1). - Bernard Schott, May 13 2022

Links

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

Flávio V. Fernandes, A306771(n) equals 3 times A007310(n)

Example

33 is in the sequence because:
phi(33) = phi(11 + 22) = phi(11) + phi(22) = 10 + 10 = 20, and
phi(33) = phi(3 * 11) = phi(3) * phi(11) = 2 * 10 = 20.

Maple

with(numtheory):
for n from 1 to 500 do:
ii:=0:
  for i from 1 to trunc(n/2) while(ii=0) do:
   if phi(i)+ phi(n-i)= phi(n) and n/i = floor(n/i)
      and phi(i)*phi(n/i)=phi(n)
      then
      ii:=1:printf(`%d, `, n):
      else
   fi:
  od:
od:

Prog

(PARI) isok(m) = {my(phim = eulerphi(m)); for (i=1, m\2, if ((eulerphi(i) + eulerphi(m-i) == phim) && !frac(m/i) && (eulerphi(m/i)*eulerphi(i) == phim), return (1)); ); } \\ Michel Marcus, Mar 09 2019

Crossrefs

Cf. A000010, A000040, A007310, A001748.

Sequence in context: A129966 A354675 A243128 * A216521 A110172 A261274

Adjacent sequences: A306768 A306769 A306770 * A306772 A306773 A306774

Keyword

nonn

Author

Michel Lagneau, Mar 09 2019

Extensions

Incorrect comment deleted by Peter Munn, May 12 2020

Name corrected by Flávio V. Fernandes, Aug 26 2021 and Peter Munn, Sep 03 2021

Status

approved