cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, May 08 2018

Keywords

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",")))

A304241 Indices where Mertens function A002321 reaches record amplitudes between zeros.

Original entry on oeis.org

1, 31, 114, 199, 443, 665, 1109, 1637, 2803, 7021, 8511, 9861, 19291, 24185, 31990, 42961, 48433, 59577, 96014, 141869, 230399, 300551, 355733, 603151, 926265, 1066854, 1793918, 3239797, 5343761, 6481601, 7109110, 10194458, 12874814, 30919091, 61913863
Offset: 1

Views

Author

M. F. Hasler, May 08 2018

Keywords

Comments

A subsequence of A051402 and A304239.
These are the indices where the Mertens function M = A002321 not only reaches a record value (in absolute value), but also its largest amplitude between subsequent zeros (as to avoid many "intermediate" records).

Links

Crossrefs

Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).
Cf. A304242, A304239, A304240.

Programs

  • PARI
    L=M=0;for(n=1,oo,if(m=A002321(n),abs(m)>abs(M)&&[M,N]=[m,n],abs(M)>abs(L) && (L=M) && print1(N",");M=0))

Extensions

More terms from Bert Dobbelaere, Oct 30 2018

A304242 Increasingly larger (in absolute value) extrema of the Mertens function A002321 between subsequent zeros.

Original entry on oeis.org

1, -4, -6, -8, -9, -12, -15, -16, -25, -29, 35, -43, 51, -72, 73, -88, 96, -113, -132, -134, -154, 240, -258, -278, -368, 432, 550, -683, -847, 1060, -1078, 1240, -1447, -2573, 2845, -3448, -4610, -6226, 6695, -8565, 9132, 10246, -15335, -17334, 21777, -25071
Offset: 1

Views

Author

M. F. Hasler, May 08 2018

Keywords

Comments

Values of A002321 at the indices listed in A304241.
These are those records of the absolute value of A002321 which are the maxima or minima between subsequent zeros. Figuratively speaking, these are the increasingly larger heights of the mountains or depths of the valleys of the graph of A002321.

Links

Crossrefs

Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).
Cf. A304241, A304239, A304240.

Programs

  • PARI
    L=M=0; for(n=1,oo, if(m=merten(n), abs(m)>abs(M) && [M,N]=[m,n], abs(M)>abs(L) && (L=M) && print1(L","); M=0))
  • PARI
    print1(j=1);for(i=1,#A051402-1,while( A028442[j] < A051402[i], j++); if( A028442[j-(j>1)]<=A051402[i] && A028442[j] < A051402[i+1], print1(","A002321(A051402[i])))) \\ Using precomputed vectors A002321 and A051402, e.g. from the b-files: {c=0;AX=apply(t->fromdigits(digits(t)[#Str(c++)+1..-1]),readvec("/tmp/bX.txt"))}

Formula

a(n) = A002321(A304241(n)).

Extensions

More terms from Bert Dobbelaere, Oct 30 2018
Showing 1-3 of 3 results.

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