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A320192
Number of summation terms of the reciprocal integer squares series that gives the best approximation to n terms of the Euler product for zeta(2).
0
3, 6, 12, 20, 27, 37, 46, 59, 72, 84, 98, 111, 125, 140, 157, 172, 188, 205, 221, 239, 258, 277, 296, 316, 334, 353, 374, 395, 418, 441, 462, 484, 505, 528, 549, 572, 595, 618, 641, 664, 688, 712, 736, 761, 786, 813, 838, 862, 886, 912, 937, 963, 991
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
Cf. A007406, A007407, A013661 (zeta(2)), A261208.
Sequence in context: A226220 A218076 A308722 * A160732 A320608 A246147
KEYWORD
nonn
AUTHOR
Horst H. Manninger, Oct 07 2018
STATUS
approved