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A060461
Numbers k such that 6*k-1 and 6*k+1 are twin composites.
12
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
1,1
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
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
EXAMPLE
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).
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 A. 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 *)
PROG
(PARI) A060461()={my(maxx=5000); n=1; ctr=0; while(ctr<maxx, if(!isprime(6*n-1)&&!isprime(6*n+1), print1(n, ", "); ctr+=1); n+=1); } \\ Bill McEachen, Apr 04 2015
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Lekraj Beedassy, Apr 09 2001
EXTENSIONS
More terms from James A. Sellers, Apr 11 2001
STATUS
approved