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%I #36 Mar 19 2017 19:33:59
%S 1,3,5,9,11,15,17,25,27,29,33,41,45,51,55,59,71,75,81,85,87,99,101,
%T 107,121,123,125,135,137,145,149,153,165,177,179,187,191,197,205,213,
%U 225,227,239,243,255,261,269,275,281,289,295,297,303,311,319,321,347,355,363,369,375,405,411,419,425,431,435,447,451,459,461,493,495,505,521,531,535,537,561,569
%N The factorization of n contains only lesser of twin primes.
%C 1 is in the sequence by convention.
%C The longest chain of consecutive odd numbers in this sequence has length 6 (otherwise one of the terms is divisible by 7). - _Zak Seidov_, Jun 20 2016
%C The first occurrence of five consecutive odd numbers is a(1473448)..a(1473452) = {80008203 = 3 * 11 * 2424491, 80008205 = 5 * 17^3 * 3257, 80008207 = 2081 * 38447, 80008209 = 3^3 * 2963267, 80008211}. - _Charles R Greathouse IV_, Jun 30 2016
%C Smallest n (if any) such that n+{0,2,4,6,8,10} all are terms is n > 10^12 according to _Giovanni Resta_. - _Zak Seidov_, Jul 01 2016
%C For any two terms, the sequence also contains their product. Reciprocally, this allows us to generate the whole sequence which is the closure, with respect to multiplication, of the set A001359 of lesser of twin primes. - _M. F. Hasler_, Jun 23 2016
%H Zak Seidov, <a href="/A274212/b274212.txt">Table of n, a(n) for n = 1..15000</a>
%F Arithmetic conjecture: the equation a(n+1) - a(n) = 2r has infinitely many solutions for any fixed integer value r >= 1.
%F Analytic conjecture: a(n) is asymptotic to C*n*log(n)^2 for a constant C > 0.2.
%o (PARI) for(n=1,1000,if(prod(i=1,omega(n),isprime(factor(n)[i,1]+2))==1,print1(n,",")))
%o (PARI) is(n)=!for(i=1,#n=factor(n)~,isprime(n[1,i]+2)||return) \\ prefix "bittest(n,0) &&" for efficiency, if the selection is to be applied to numbers of unknown parity. - _M. F. Hasler_, Jun 23 2016
%o (PARI) list(lim,mx=lim)=my(u,v=List([1]),P=List(),p=2); forprime(q=3, min(mx, lim)+2, if(q-p==2, listput(P,p)); p=q); for(i=1,#P, p=P[i]; if(3*p>lim, for(j=i,#P, listput(v,P[j])); break); u=list(lim\p,p); for(j=1,#u, listput(v,p*u[j]))); Set(v) \\ _Charles R Greathouse IV_, Jun 30 2016
%Y Cf. A001359.
%K nonn
%O 1,2
%A _Benoit Cloitre_, Jun 13 2016
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