A304239 Indices for which the Mertens function A002321 reaches its extremum between subsequent zeros for the first time.

1, 31, 43, 61, 73, 95, 114, 146, 154, 161, 165, 199, 221, 233, 237, 246, 286, 330, 341, 354, 357, 359, 365, 374, 395, 402, 406, 410, 417, 421, 426, 443, 538, 586, 619, 665, 782, 787, 794, 797, 803, 813, 818, 830, 851, 861, 871, 879, 885, 887, 890, 894, 897, 901, 905, 907, 911

Offset: 1

Author

M. F. Hasler, May 08 2018

Keywords

• nonn

Comments

This is related to the Mertens conjecture, more precisely to record values of Mertens function A002321 in the following sense: Due to the short-scale and long-scale oscillations of A002321, it is less appealing to consider record values in the usual sense (cf. A051402), which yields many slowly growing records and record indices lying closely together, during the approach of a "long-scale" record. Therefore this sequence considers maxima or minima between two subsequent zeros, ignoring the empty intervals between immediately adjacent zeros A002321(k) = A002321(k+1) = 0.

The values of these extrema are listed in A304240(n) = A002321(A304239(n)).

Then one can consider the sequence of indices where the corresponding values of A002321 have opposite sign, and/or are larger in absolute value than the preceding record amplitude in the above sense, cf. A304240 & A304241: These are the points which one would really consider as record maxima / minima when looking at the graph on a larger scale.

Examples

			The initial value a(1) = 1 may be considered conventional, or the maximum reached between M(0) = 0 (empty sum) and M(2) = 0, where we write M for the Mertens function A002321.
After M(2) = 0, Mertens's function has negative values up to the next zero, M(39) = 0. The largest negative value is -4 = M(31) = M(32). Therefore a(2) = 31.
Since M(39) = M(40) = 0, the maximum amplitude between these two consecutive zeros would be zero, and is ignored by definition.
The next "local minimum" of this type is reached at M(43) = -3, this value is taken several times up to the next zero at n = 58. Therefore a(3) = 43.
The next such "local minima" are M(61) = -2 and M(73) = -4, so a(4) = 61, a(5) = 73.
It is only at n = 94 that M takes a positive value for the first time after M(1) = 1, and M(95) = 2 is the largest value reached until the next zero (at n = 101), so a(6) = 95.
And so on.
		

Crossrefs

Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).

Cf. A304240 (values of the extrema), A304241 (indices of increasingly larger extrema), A304242 (the associated values).

Programs

PARI

M=0; for(n=1,oo, if(m=A002321(n),abs(m)>abs(M)&& [M,N]=[m,n], M&& M=printf(N",")))