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.

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).

Links

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

David Morales Marciel, Euler's Totient function statement proposal

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

Adjacent sequences: A256238 A256239 A256240 * A256242 A256243 A256244

Keyword

nonn

Author

David Morales Marciel, Apr 19 2015

Extensions

Removed long Python code, and added very simple Python program (two lines) with sympy as suggested by the Editor.

Status

approved