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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.
2
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
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
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