A073831 -id:A073831 - cp's OEIS frontend

A073830 a(n) = 4((n-1)! + 1) + n (mod n(n + 2)).

0, 2, 0, 8, 0, 10, 56, 12, 22, 14, 0, 16, 182, 18, 34, 20, 0, 22, 380, 24, 46, 26, 552, 28, 29, 30, 58, 32, 0, 34, 992, 36, 37, 38, 74, 40, 1406, 42, 82, 44, 0, 46, 1892, 48, 94, 50, 2256, 52, 53, 54, 106, 56, 2862, 58, 59, 60, 118, 62, 0, 64, 3782, 66, 67, 68, 134, 70

Offset: 1

Reinhard Zumkeller, Aug 12 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell, Twin Prime
P. A. Clement, Congruences for sets of primes, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 23-25.
PlanetMath, Clement’s theorem on twin primes.

Cf. A001359, A005563, A006512, A073831, A073832.

[(4*(Factorial(n-1)+1)+n) mod (n^2+2*n): n in [1..70]]; // Vincenzo Librandi, May 04 2014

A073830 := proc(n)
    4*((n-1)!+1)+n ;
    modp(%,n*(n+2)) ;
end proc:
seq(A073830(n),n=1..60) ; # R. J. Mathar, Feb 21 2017

a[n_] := Mod[4 ((n - 1)! + 1) + n, n (n + 2)]; Array[a, 66] (* Jean-François Alcover, Feb 22 2018 *)

a(n) = A073829(n) mod A005563(n).
For n > 1: a(n) = 0 iff (n, n+2) are twin primes (Clement, 1949).
From Bernard Schott, Nov 16 2021: (Start)
If p is an odd prime, and p+2 is composite, then a(p) = p*(p+1).
If m is composite, and m+2 is prime, then a(m) = 2*(m+2).
If n even >= 4, a(n) = n + 4.
If p prime >= 5, a(p^2) = p^2 + 4. (End)

A073832 k between A001359(n) and A001359(n+1) such that A073830(k) is maximal.

Original entry on oeis.org

4, 7, 13, 23, 37, 53, 67, 97, 103, 131, 139, 173, 181, 193, 223, 233, 263, 277, 307, 337, 409, 421, 457, 509, 563, 593, 613, 631, 653, 797, 811, 823, 853, 877, 1013, 1021, 1039, 1051, 1087, 1129, 1223, 1259, 1283, 1297, 1307, 1423, 1447, 1471, 1483, 1601

Offset: 1

Reinhard Zumkeller, Aug 12 2002

nonn

A073830(a(n)) = A073831(n).

Michael S. Branicky, Table of n, a(n) for n = 1..2600

A073832 := proc(n)
    local k,kmx,a ;
    kmx := 0 ;
    a := A001359(n)+1 ;
    for k from A001359(n)+1 to A001359(n+1)-1 do
        if A073830(k) > kmx then
            a := k ;
            kmx := A073830(k) ;
        end if;
    end do:
    a ;
end proc:
seq(A073832(n),n=1..50) ; # R. J. Mathar, Feb 21 2017

f[n_] := Mod[4*((n - 1)! + 1) + n, n*(n + 2)];
pp = Select[Prime[Range[300]], PrimeQ[# + 2] & ];
a[n_] := MaximalBy[Range[pp[[n]], pp[[n + 1]]], f];
Array[a, Length[pp] - 1] // Flatten (* Jean-François Alcover, Feb 22 2018 *)

from math import factorial
from itertools import islice, pairwise
from sympy import isprime, nextprime, primerange
def f(n): return (4*(factorial(n-1) + 1) + n)%(n*(n + 2))
def bgen(): # generator of A001359
    p, q = 2, 3
    while True:
        if q - p == 2: yield p
        p, q = q, nextprime(q)
def agen(): # generator of terms
    for p, q in pairwise(bgen()):
        yield max((f(k), k) for k in range(p+1, q))[1]
print(list(islice(agen(), 80))) # Michael S. Branicky, Aug 13 2024

cp's OEIS Frontend

A073830 a(n) = 4((n-1)! + 1) + n (mod n(n + 2)).

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A073832 k between A001359(n) and A001359(n+1) such that A073830(k) is maximal.

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cp's OEIS Frontend

A073830 a(n) = 4*((n-1)! + 1) + n (mod n*(n + 2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A073832 k between A001359(n) and A001359(n+1) such that A073830(k) is maximal.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

Python

A073830 a(n) = 4((n-1)! + 1) + n (mod n(n + 2)).