A355626 -id:A355626 - cp's OEIS frontend

A355243 a(n) is the largest integer value of Product_{k=1..n} (2 + 1/t_k) with integers t_k > 1.

5, 11, 25, 55, 125, 277, 637, 1421, 3237, 7553, 16807, 38661, 90209, 208397

Offset: 2

Hugo Pfoertner and Markus Sigg, Jul 16 2022

			a(2) = 5 = (2 + 1/3) * (2 + 1/7),
a(3) = 11 = (2 + 1/3) * (2 + 1/5) * (2+1/7) = (2 + 1/3)^2 * (2 + 1/49),
a(7) = 277 = (2 + 1/3)^5 * ( 2 + 1/405) * (2 + 1/13630477).

A355626 gives more information.

A355630 same problem with factors (3 + 1/t_k).

PARI
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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.

Original entry on oeis.org

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,", ")));

A334127 Number of nonempty sets {p_1, p_2, ..., p_k} such that Product_{i=1..k} p_i divides Product_{i=1..k} (n + p_i), where the p_i are distinct primes.

Original entry on oeis.org

1, 3, 4, 7, 6, 19, 8, 17, 8, 25, 12, 105, 8, 35, 22, 24, 16, 59, 28, 77, 30, 26, 22, 159, 8, 117, 23, 161, 26, 787, 32, 69, 46, 57, 30, 534, 32, 69, 90, 137, 22, 707, 20, 266, 54, 73, 50, 423, 38, 626, 62, 229, 52, 1324, 220, 489, 130, 285, 58, 2943, 24, 119, 274, 171, 202, 12089, 46, 181, 158, 201, 66, 1999, 58, 391, 234, 917, 126, 451, 42, 1767, 73, 1034, 86, 34691, 81, 150, 142, 233, 94, 18319, 226, 477, 70, 477, 114, 4419, 54, 157, 234, 2252

Offset: 1

Jinyuan Wang, May 10 2020

nonn

a(n) is always finite. Proof: let p_1 < p_2 < ... < p_k, we can prove p_k <= 2*n^2 + n. If p_k > 2*n^2 + n, then 2*p_k > p_k + n, thus p_k - n is in the set. If p_k - m*n is in the set and m < n, then 2*(p_k - m*n) > p_k + n, thus p_k - (m+1)*n is in the set. Therefore, p_k - m*n are in the set for 0 <= m <= n. Since p_k - n*n > n + 1, p_k - m*n can be divisible by n + 1 for some m <= n, which is a contradiction to the p_i being primes.

			For n = 3, {3}, {2, 3}, {2, 5} and {2, 3, 5} are such sets, thus a(3) = 4.

Cf. A085098, A118086, A355626, A355627, A355628, A355629, A355630, A355631, A376011, A376012,

a(n, k=primepi(2*n^2+n)) = {my(c=-1, p=primes(k)); forsubset(k, v, if(vecprod(vector(#v, i, p[v[i]]+n))%vecprod(vector(#v, i, p[v[i]])) == 0, c++)); c; }

Terms a(13) onward from Max Alekseyev, Apr 08 2025

cp's OEIS Frontend

A355243 a(n) is the largest integer value of Product_{k=1..n} (2 + 1/t_k) with integers t_k > 1.

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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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A334127 Number of nonempty sets {p_1, p_2, ..., p_k} such that Product_{i=1..k} p_i divides Product_{i=1..k} (n + p_i), where the p_i are distinct primes.

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