OFFSET
1,1
COMMENTS
Conjecture: It appears that for integer n the number of summation terms of the reciprocal integer squares series that gives the best approximation to Pi^2/6 - 1/n is n. - Andrew Howroyd, Nov 24 2018
FORMULA
Conjecture: a(n) = floor(1/(Pi^2/6 - Product{k=1..n} 1/(1-1/prime(k)^2) )). - Andrew Howroyd, Nov 24 2018
EXAMPLE
First term of Euler product: 4/3 needs 3 terms for best approximation: 1 + 1/4 + 1/9.
Case n=2: The second term of the Euler product is 1/((1-1/2^2)*(1-1/3^2)) = 3/2 = 1.5 which lies between Sum_{k=1..6} 1/k^2 and Sum_{k=1..7} 1/k^2. The first of these is the better approximation so a(2) = 6.
MATHEMATICA
x = 70;
y = Round[x^1.8];
eulerp = Table[Product[1./(1 - 1/Prime[n]^2), {n, 1, k}], {k, 1, y}];
sums = Table[{k, Sum[1./n^2, {n, 1, k}]}, {k, 1, y}];
diff = Table[Abs[eulerp[[r]] - sums[[All, 2]]], {r, x}];
count = Flatten[Array[Range[y] &, x]];
fldiff = Flatten[diff];
t5 = Transpose[{count, fldiff}];
t6 = Partition[t5, y];
t7 = Table[MinimalBy[t6[[i]], Last], {i, x}];
sol = Flatten[t7, 1][[All, 1]]
PROG
(PARI) \\ to make this faster use floating point, but beware precision.
c(r)={my(s=r-r, k=0); while(s < r, k++; s+=1/k^2); k - (2*(s-r) >= 1/k^2)}
a(n)={c(prod(k=1, n, 1/(1-1/prime(k)^2)))} \\ Andrew Howroyd, Nov 23 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Horst H. Manninger, Oct 07 2018
STATUS
approved