A060461 Numbers k such that 6*k-1 and 6*k+1 are twin composites.

20, 24, 31, 34, 36, 41, 48, 50, 54, 57, 69, 71, 79, 86, 88, 89, 92, 97, 104, 106, 111, 116, 119, 130, 132, 134, 136, 139, 141, 145, 149, 150, 154, 160, 167, 171, 174, 176, 179, 180, 189, 190, 191, 193, 196, 201, 207, 209, 211, 212, 219, 222, 223, 224, 225, 226

Offset: 1

Author

Lekraj Beedassy, Apr 09 2001

Keywords

• nonn

Comments

A counterpart to A002822, which generates twin primes.

Intersection of A046953 and A046954. - Michel Marcus, Sep 27 2013

All terms can be expressed as (6ab+a+b OR 6cd-c-d) AND (6xy+x-y) for a,b,c,d,x,y positive integers. Example: 20=6*2*2-2-2 AND 20=6*3*1+3-1. - Pedro Caceres, Apr 21 2019

Examples

			a(9) = 57: the 9th twin composites among the odds are { 6*57-1, 6*57+1 }, i.e., (341, 343) or (11*31, 7^3).
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A002822, A046953, A046954, A259826.

Programs

MATLAB

i=1:10000;
Q1 = 6*i-1;
Q2 = 6*i+1;
Q = union(Q1,Q2);
P = primes(max(Q));
AT = setxor(Q,P);
f = 0;
for j=1:numel(AT);
    K = AT(j);
    K2 = K+2;
    z = ismember(K2,AT);
    if z == 1;
        f = f+1;
        ATR(f,:) = K + 1;
    end
end
m6 = ATR./6;
% Jesse H. Crotts, Sep 05 2016

Maple

iscomp := proc(n) if n=1 or isprime(n) then RETURN(0) else RETURN(1) fi: end: for n from 1 to 500 do if iscomp(6*n-1)=1 and iscomp(6*n+1)=1 then printf(`%d,`,n) fi: od: # James Sellers, Apr 11 2001

Mathematica

Select[Range[200], !PrimeQ[6#-1]&&!PrimeQ[6#+1]&] (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Select[Range[300],AllTrue[6#+{1,-1},CompositeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 15 2015 *)
Select[Range@ 300, Times @@ Boole@ Map[CompositeQ, 6 # + {1, -1}] > 0 &] (* Michael De Vlieger, Sep 14 2016 *)

PARI

A060461()={my(maxx=5000); n=1; ctr=0; while(ctrBill McEachen, Apr 04 2015

Python

from sympy import isprime; from sys import maxsize as oo
is_A060461 = lambda n: not (isprime(n*6-1) or isprime(n*6+1))
def A060461(n = None, first = oo, start = 1, end = oo):
    "Return the n-th term or a generator of up to 'first' terms less than 'end', starting at 'start'."
    if n: first = n
    seq = (m for m,_ in zip(filter(is_A060461, range(start,end)), range(first)))
    return max(seq) if n else seq
list(A060461(first=20)) # M. F. Hasler, Jul 10 2025

Formula

a(n) ~ n. More specifically, there are x - x/log x + O(x/log^2 x) terms of the sequence up to x. - Charles R Greathouse IV, Mar 03 2020

a(n) = A259826(n)/6. - M. F. Hasler, Jul 10 2025

Extensions

More terms from James Sellers, Apr 11 2001