A356211 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.

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

Offset: 1

Author

Hugo Pfoertner and Markus Sigg, Aug 16 2022

Keywords

• nonn
• more

Comments

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

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.

Examples

			1 is not a term because the empty product has the value 1.
Other odd numbers that are not terms:
    5 = (2 + 1/3) * (2 + 1/7);
    9 = (2 + 1/9) * (2 + 1/ 13) * (2 + 1/19);
   11 = (2 + 1/3) * (2 + 1/5) * (2 + 1/7);
   17 = (2 + 1/25) * (2 + 1/27) * (2 + 1/37) * (2 + 1/55);
  255 = (2 + 1/3)^4 * (2 + 1/7) * (2 + 1/139) * (2 + 1/10633).
		

Crossrefs

Cf. A355243, A355516, A355626.

Programs

PARI

\\ Using the function nTuples from the linked file in A355626 and setting the global variable s:
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,", ")));