A355243 -id:A355243 - cp's OEIS frontend

A355516 a(n) is the number of distinct integer values of Product_{k=1..n} (2 + 1/t_k) with integers t_k > 1.

1, 2, 5, 11, 29, 70, 164, 392, 933

Offset: 2

Hugo Pfoertner and Markus Sigg, Jul 16 2022

			a(2) = 1: The only representable integer is 5 = (2 + 1/3) * (2 + 1/7).
a(3) = 2: 9 and 11 can be represented; 9 = (2 + 1/9) * (2 + 1/13) * (2 + 1/19) and 10 other ways; 11 = (2 + 1/3)*(2 + 1/5)*(2 + 1/7) = (2 + 1/3)^2 * (2 + 1/49).

A355626 provides more information.
A355628 same problem with factors (3 + 1/t_k).
Cf. A355243.

PARI
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\\ See link in A355626.
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A356210 a(n) is the number of tuples (t_1, ..., t_n) with integers 2 <= t_1 <= ... <= t_n such that 2^n + 1 = Product_{i = 1..n} (2 + 1/t_i).

Original entry on oeis.org

0, 1, 11, 430, 364693

Offset: 1

Hugo Pfoertner and Markus Sigg, Aug 27 2022

			a(1) = 0 trivially;
a(2) = 1 because the only way to express 2^2 + 1 = 5 is (2 + 1/3)*(2 + 1/7);
a(3) = 11: the lexicographically earliest tuple is (5, 23, 517), and the lexicographically latest tuple is (9, 13, 19);
a(4) = 430: lexicographically earliest is (9, 77, 5891, 34700935), lexicographically latest is (25, 27, 37, 55);
a(5) = 364693: lexicographically earliest is (17, 281, 78821, 6212710631, 38597773381434062845), lexicographically latest is (57, 77, 85, 93, 115).

A355626 provides more information.
A355629 is the same problem with target 3^n + 1 and factors (3 + 1/t_k).
Cf. A355243, A355516, A356211.

\\ see link in A355626; set s=2 and use function a355629(n).

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.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner and Markus Sigg, Aug 16 2022

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.

			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).

Cf. A355243, A355516, A355626.

\\ 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,", ")));

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A355516 a(n) is the number of distinct integer values of Product_{k=1..n} (2 + 1/t_k) with integers t_k > 1.

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A356210 a(n) is the number of tuples (t_1, ..., t_n) with integers 2 <= t_1 <= ... <= t_n such that 2^n + 1 = Product_{i = 1..n} (2 + 1/t_i).

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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.

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