A234800 - OEIS

%I #32 Mar 07 2023 16:22:13

%S 1,14,111,5826,85865,4580009,290820868

%N First occurrence of n in A234323: Number of nontrivial zeros of the Riemann zeta function in the interval 1/2 + i[n,n+1).

%C k

%C 0: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, ..., A120401

%C 1: 14, 21, 25, 30, 32, 37, 40, 43, 48, 49, 52, 56, 59, 60, 65, ..., A234802

%C 2: 111, 150, 169, 224, 231, 329, 357, 373, 415, 478, 493, 540, ..., A234803

%C 3: 5826, 5978, 6494, 7563, 8106, 8942, 9601, 9856, 9976, 10000, ..., A234804

%C 4: 85865, 193997, 245986, 276125, 283519, 297624, 298486, 311014, ..., A234805

%C 5: 4580009, 7149902, 8120618, 10002309, 10597386, 11333337, 11432756, ..., A234806

%C 6: 290820868, 317905108, 334924359, 386701579, 410462993, 430633085, ..., A234807

%C The occurrence of <0> is probably finite since the average height of a(n) is A234323 is log(n)/(2*Pi). This is a conjecture.

%H Simon Plouffe, <a href="/A234323/b234323.txt">Table for A234323(n) for n = 1..1000000</a>

%H LMFDB David Platt's <a href="https://www.lmfdb.org/zeros/zeta/">Table for A234323(n) of the first 103 billion zeros</a>.

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/OEIS/b234323.txt.gz">Table for A234323(n) up to 2 billion</a> - note the file is 25 gigabytes uncompressed

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function_zeros">Riemann zeta function zeros</a>

%t t = Table[0, {100}]; k = 1; cnt = 1; a = 0; t[[1]] = 14; While[k < 1000001, b = Floor[ Im[ N[ ZetaZero[ k]] ]]; If[b == a, cnt++; If[t[[cnt]] == 0, t[[cnt]] = b; Print[{cnt, b}]], cnt = 1]; a = b; k++]

%Y Cf. A234323, A002410, A122526.

%K nonn,hard,more

%O 0,2

%A _Simon Plouffe_ and _Robert G. Wilson v_, Dec 30 2013