David Morales Marciel - cp's OEIS frontend

User: David Morales Marciel

David Morales Marciel has authored 2 sequences.

A257886 Least positive m such that floor(n! / (2*(floor(n/2)!))) + m is prime.

2, 1, 2, 1, 1, 1, 1, 13, 1, 1, 29, 1, 1, 37, 29, 17, 31, 71, 71, 37, 23, 1, 37, 1, 41, 41, 31, 31, 59, 31, 41, 41, 41, 41, 41, 37, 41, 193, 83, 41, 53, 67, 149, 97, 59, 73, 113, 107, 137, 59, 137, 67, 101, 83, 73, 101, 241, 71, 73, 79, 83, 227, 199, 223, 127, 83, 83, 181, 227, 149, 103, 1, 587, 179, 229, 167, 127, 163, 109, 83

Offset: 1

David Morales Marciel, May 11 2015

nonn

Conjecture: No term is composite (similar conjecture to A033932 for a different expression).

			n = 1, floor(1! / (2*(floor(1/2)!)))=0, m = 2, and 0+2=2 is prime.
n = 2, floor(2! / (2*(floor(2/2)!)))=1, m = 1, and 1+1=2 is prime.
...
n = 15, floor(15! / (2*(floor(15/2)!)))=129729600, m = 29, and 129729600+29 = 129729629 is prime.

David Morales Marciel, About Fortunate numbers and other similar expressions

Cf. A033932.

lpm[n_]:=Module[{c=Floor[n!/(2Floor[n/2]!)]},NextPrime[c]-c]; Array[lpm,80] (* Harvey P. Dale, May 15 2018 *)

from sympy import factorial, nextprime
[(nextprime(int(factorial(n)/(2*factorial(n//2)))))-int(factorial(n)/(2*factorial(n//2))) for n in range(1,10**5)]

Edited. Wolfdieter Lang, Jun 08 2015

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.

Original entry on oeis.org

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

David Morales Marciel, Apr 19 2015

nonn

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).

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
David Morales Marciel, Euler's Totient function statement proposal

Cf. A000010.

[k:k in [1..370]| IsIntegral(k/(EulerPhi(k)/2+1))]; // Marius A. Burtea, Nov 06 2019

aQ[n_] := Divisible[n, 1 + EulerPhi[n] / 2]; Select[Range[400], aQ] (* Amiram Eldar, Nov 06 2019 *)

isok(n) = (n % (eulerphi(n)/2+1)) == 0; \\ Michel Marcus, Apr 20 2015

from sympy import totient
[n for n in range(1, 10**5) if n%((totient(n)/2)+1)==0]

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

cp's OEIS Frontend

User: David Morales Marciel

A257886 Least positive m such that floor(n! / (2*(floor(n/2)!))) + m is prime.

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