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
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).
A355629 is the same problem with target 3^n + 1 and factors (3 + 1/t_k).
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
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).
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\\ 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,", ")));
Showing 1-2 of 2 results.
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