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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.

Wikipedia, Proof of the Euler product formula for the Riemann zeta function.

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.)