A084588 Least positive integers, all distinct, that satisfy Sum_{n>0} 1/a(n)^z = 0, where z is the first nontrivial zero of the Riemann zeta function: z = (1/2 + i*y) with y=14.13472514173469379045...

1, 2, 3, 4, 5, 6, 11, 13, 16, 20, 25, 30, 36, 44, 54, 65, 78, 93, 110, 130, 153, 178, 205, 234, 266, 300, 337, 376, 418, 462, 509, 559, 611, 666, 723, 783, 845, 910, 978, 1048, 1122, 1198, 1277, 1359, 1444, 1532, 1623, 1717, 1814, 1914, 2017, 2123, 2232, 2344

Offset

1,2

Comments

Sequence satisfies: Sum_{n>0} 1/a(n)^z = 0 by requiring that the modulus of the successive partial sums are monotonically decreasing in magnitude to zero for the given z.

Sequences A084588 - A084593 are related to zeros of the Riemann zeta function. The least integers that satisfy Sum_{n>0} 1/a(n)^z = 0, where a(1)=1, a(n+1) > a(n) and z is a nontrivial zero of the Riemann zeta function.

Links

Table of n, a(n) for n=1..54.

Andrew M. Odlyzko, The first 100 (nontrivial) zeros of the Riemann Zeta function.

Index entries for zeta function.

Mathematica

Reap[For[z = ZetaZero[1]; S = 0; w = 1; a = 0; n = 1, n <= 100, n++, b = a + 1; While[Abs[S + Exp[-z*Log[b]]] > w, b++]; S = S + Exp[-z*Log[b]]; w = Abs[S]; a = b; Print[b]; Sow[b]]][[2, 1]] (* Jean-François Alcover, Oct 22 2019, from PARI *)

Prog

(PARI) S=0; w=1; a=0; for(n=1, 100, b=a+1; while(abs(S+exp(-z*log(b)))>w, b++); S=S+exp(-z*log(b)); w=abs(S); a=b; print1(b, ", "))

Crossrefs

Cf. A084589, A084590, A084591, A084592, A084593.

Sequence in context: A348256 A082657 A108378 * A309548 A088411 A075073

Adjacent sequences: A084585 A084586 A084587 * A084589 A084590 A084591

Keyword

nonn

Author

Paul D. Hanna, Jun 03 2003

Status

approved