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%I #22 Apr 16 2022 22:46:18
%S 1,2,3,4,5,6,11,13,16,20,25,30,36,44,54,65,78,93,110,130,153,178,205,
%T 234,266,300,337,376,418,462,509,559,611,666,723,783,845,910,978,1048,
%U 1122,1198,1277,1359,1444,1532,1623,1717,1814,1914,2017,2123,2232,2344
%N 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...
%C 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.
%C 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.
%H Andrew M. Odlyzko, <a href="http://www.plouffe.fr/simon/constants/zeta100.html">The first 100 (nontrivial) zeros of the Riemann Zeta function.</a>
%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%t 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 *)
%o (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,","))
%Y Cf. A084589, A084590, A084591, A084592, A084593.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 03 2003
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