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Odd numbers which are neither of the form p*2^m + 1 nor of the form p*2^m - 1 with p prime.
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%I #33 Sep 08 2022 08:46:09

%S 71,99,101,109,131,139,155,169,181,197,199,221,229,239,241,251,259,

%T 265,281,287,289,307,309,311,323,337,339,341,349,365,371,373,379,391,

%U 401,407,409,419,431,433,439,441,443,461,469,475,485,491,493,499,505,517,519

%N Odd numbers which are neither of the form p*2^m + 1 nor of the form p*2^m - 1 with p prime.

%C For each n, the sequence has a set of n consecutive odd numbers.

%C For any n, the number 2*A140077(n) + 1 is in the sequence.

%C Every number of the form S*2^n + 1 or R*2^n - 1 with n > 0, where S is a composite SierpiƄski number and R is a composite Riesel number, is in the sequence.

%C Odd numbers n such that (n-1)/A007814(n-1) and (n+1)/A007814(n+1) are composite. - _Robert Israel_, Nov 19 2014

%p filter:= proc(n)

%p local m1,m2;

%p m1:= padic[ordp](n-1,2);

%p if n-1 = 2^m1 then return false fi;

%p m2:= padic[ordp](n+1,2);

%p n+1 <> 2^m2 and not isprime((n-1)/2^m1) and not isprime((n+1)/2^m2);

%p end proc:

%p select(filter, [seq(2*i+1,i=0..1000)]); # _Robert Israel_, Nov 19 2014

%o (Magma) lst1:=[]; lst2:=[]; r:=519; t:=Floor(Log(2, r))-1; for m in [0..t] do e:=Floor(r/2^m); for p in [2..e] do if IsPrime(p) then a:=p*2^m-1; b:=p*2^m+1; if not a in lst1 then Append(~lst1, a); end if; if not b in lst1 then Append(~lst1, b); end if; end if; end for; end for; for n in [3..r by 2] do if not n in lst1 then Append(~lst2, n); end if; end for; lst2;

%o (PARI) b=0; forstep(n=1, 519, 2, c=2^floor(log(n)/log(2)); a=b; b=(n+1)/gcd(n+1, c); if(a>8&&!isprime(a)&&!isprime(b), print1(n, ", ")));

%Y Cf. A007814, A140077.

%K nonn

%O 1,1

%A _Arkadiusz Wesolowski_, Nov 18 2014