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A256241
Numbers n whose Euler's totient function phi(n), divided by two, plus one, p = (phi(n) / 2) + 1, is a divisor of n.
1
4, 6, 12, 15, 20, 21, 28, 30, 33, 39, 42, 44, 51, 52, 57, 66, 68, 69, 76, 78, 87, 92, 93, 102, 111, 114, 116, 123, 124, 129, 138, 141, 148, 159, 164, 172, 174, 177, 183, 186, 188, 201, 212, 213, 219, 222, 236, 237, 244, 246, 249, 258, 267, 268, 282, 284, 291, 292, 303, 309, 316, 318, 321, 327, 332, 339, 354, 356, 366
OFFSET
1,1
COMMENTS
p is always a prime factor of n as well.
Except for the case n=6, p is always the greatest prime factor of n.
(n/p) is an upper bound on the rest of the prime factors 'q' of n, so always q <= (n/p).
EXAMPLE
For n = 4, phi(n) = 2, p = (phi(n)/2)+1 = 2, is prime and is a prime factor of 4.
For n = 6, phi(n) = 2, p = (phi(n)/2)+1 = 2, is prime and is a prime factor of 6.
MATHEMATICA
aQ[n_] := Divisible[n, 1 + EulerPhi[n] / 2]; Select[Range[400], aQ] (* Amiram Eldar, Nov 06 2019 *)
PROG
(Python)
from sympy import totient
[n for n in range(1, 10**5) if n%((totient(n)/2)+1)==0]
(PARI) isok(n) = (n % (eulerphi(n)/2+1)) == 0; \\ Michel Marcus, Apr 20 2015
(Magma) [k:k in [1..370]| IsIntegral(k/(EulerPhi(k)/2+1))]; // Marius A. Burtea, Nov 06 2019
CROSSREFS
Cf. A000010.
Sequence in context: A131863 A074870 A251630 * A364385 A247632 A104236
KEYWORD
nonn
AUTHOR
EXTENSIONS
Removed long Python code, and added very simple Python program (two lines) with sympy as suggested by the Editor.
STATUS
approved