A356211 - OEIS

%I #12 Aug 24 2022 09:23:16

%S 3,7,13,15,27,29,31,53,57,59,61,63,107,123,127

%N Odd numbers that cannot be written as a product of an arbitrary number of rational factors of the form 2 + 1/t_k with integers t_k > 1.

%C It is conjectured that there are no further terms. This was checked up to 2^21.

%C If x > 3 is an element of the sequence and y := (x-1)/2 is odd, then y is an element of the sequence. Because if y > 1 is a product of n factors (2 + 1/t_k) with integers t_k > 1, then x = 2*y + 1 = y * (2 + 1/y) is a product of n+1 such factors.

%e 1 is not a term because the empty product has the value 1.

%e Other odd numbers that are not terms:

%e     5 = (2 + 1/3) * (2 + 1/7);

%e     9 = (2 + 1/9) * (2 + 1/ 13) * (2 + 1/19);

%e    11 = (2 + 1/3) * (2 + 1/5) * (2 + 1/7);

%e    17 = (2 + 1/25) * (2 + 1/27) * (2 + 1/37) * (2 + 1/55);

%e   255 = (2 + 1/3)^4 * (2 + 1/7) * (2 + 1/139) * (2 + 1/10633).

%o (PARI) \\ Using the function nTuples from the linked file in A355626 and setting the global variable s:

%o s = 2; L = vector(3815); for (n = 2, 9, forstep (k = 2^n+1, (5/2)^n, 2, my (istup=nTuples(n,k,1,0)); if(istup, L[k]++))); forstep (k=2^10+1, 2^11-1, 2, my (istup=nTuples(10,k,1,0)); if(istup, L[k]++)); forstep (k=3, 2048, 2, if(L[k]==0, print1(k,", ")));

%Y Cf. A355243, A355516, A355626.

%K nonn,more

%O 1,1

%A _Hugo Pfoertner_ and _Markus Sigg_, Aug 16 2022