

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
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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 * A247632 A104236 A265225
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



