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Least positive integers, all distinct, that satisfy Sum_{n>0} 1/a(n)^z = 0, where z is the fifth nontrivial zero of the Riemann zeta function: z=(1/2 + i*y) with y=32.935061587739189690662368964...
6

%I #13 Apr 16 2022 22:47:06

%S 1,2,4,5,20,58,64,84,91,99,108,118,129,142,156,170,185,201,219,238,

%T 257,277,299,323,348,374,402,432,463,495,529,566,606,649,695,744,796,

%U 851,909,969,1031,1095,1162,1232,1305,1381,1459,1540,1623,1709,1797,1888

%N Least positive integers, all distinct, that satisfy Sum_{n>0} 1/a(n)^z = 0, where z is the fifth nontrivial zero of the Riemann zeta function: z=(1/2 + i*y) with y=32.935061587739189690662368964...

%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.

%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>

%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. A084588, A084589, A084591, A084592, A084593.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 04 2003