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 A328655 Least k such that Sum_{m=1..k} 1/m^2 > Product_{i=1..n} 1/(1 - 1/prime(i)^2). 2
 3, 7, 12, 20, 27, 37, 47, 59, 72, 84, 98, 112, 125, 141, 157, 173, 188, 205, 222, 239, 258, 277, 297, 316, 335, 354, 374, 395, 418, 442, 463, 484, 506, 528, 550, 573, 595, 618, 641, 665, 688, 713, 737, 761, 787, 813, 838, 862, 887, 912, 938, 964, 991, 1017, 1044, 1070, 1097, 1125, 1152, 1181 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Euler proved that for every s > 1, Sum_{m>=1} 1/m^s = Product_{n>=1} 1/(1 - 1/prime(n)^s) = zeta(s). This sequence compares partial sums with partial products for the case s = 2. For the case s = 1 and partial sums of harmonic series and Eulers partial products see A328684. LINKS Robert Israel, Table of n, a(n) for n = 1..3000 Leonhard Euler, Variae observationes circa series infinitas, (Various Observations about Infinite Series), published by St Petersburg Academy, 1737. EXAMPLE a(2)=7 because Product_{n=1..2} 1/(1 - 1/prime(n)^2) < Sum_{m=1..7} 1/m^2, since 3/2 = 1.5 < 266681/176400 = 1.5118... MAPLE N:= 100: # for a(1)..a(N) p:= 1: P:= 1: for n from 1 to N do   p:= nextprime(p);   P:= P * 1/(1-1/p^2);   PP[n]:= P; od: k:= 1: S:= 0: notdone:= true: for n from 1 while notdone do   S:= S + 1/n^2;   while S > PP[k] do     A[k]:= n;     k:= k+1;     if k > N then notdone:= false; break fi   od od: seq(A[i], i=1..N); # Robert Israel, Dec 10 2019 MATHEMATICA dd = {}; h = 1; hh = 1; k = 1; m = 1; Do[k = k (1/(1 - Prime[n]^-2)); kk = N[k, 30]; While[kk > hh, h = h + 1/(m + 1)^2; hh = N[h, 30]; m++]; AppendTo[dd, m], {n, 1, 68}]; dd PROG (PARI) a(n) = my(k=1, pp = prod(i=1, n, 1/(1 - 1/prime(i)^2)), s = 1); while (s <= pp, k++; s += 1/k^2); k; \\ Michel Marcus, Oct 29 2019 (PARI) apply( {A328655(n, s=2, p=1/prod(k=1, n, 1-prime(k)^-s))=for(k=1, oo, (p-=k^-s)<0&&return(k))}, [1..60]) \\ optional 2nd arg allows to compute analog for other powers s (float avoids exact calculation using fractions, use with care). - M. F. Hasler, Oct 31 2019 CROSSREFS Cf. A328684. Sequence in context: A011899 A002498 A172115 * A091369 A036698 A279169 Adjacent sequences:  A328652 A328653 A328654 * A328656 A328657 A328658 KEYWORD nonn AUTHOR Artur Jasinski, Oct 24 2019 STATUS approved

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Last modified April 3 14:34 EDT 2020. Contains 333197 sequences. (Running on oeis4.)