A362082 Numbers k achieving record deficiency via a residue-based measure, M(k) = (k+1)*(1 - zeta(2)/2) - 1 - ( Sum_{j=1..k} k mod j )/k.

1, 5, 11, 23, 47, 59, 167, 179, 359, 503, 719, 1439, 5039, 6719, 7559, 15119, 20159, 52919, 75599, 83159, 166319, 415799, 720719, 831599, 1081079, 2162159, 4324319, 5266799, 7900199, 10533599, 18345599, 28274399, 41081039, 136936799, 205405199, 410810399

Offset

1,2

Comments

M(k) = (k+1)*(1 - zeta(2)/2) - 1 - ( Sum_{j=1..k} k mod j )/k is a measure of either abundance (sigma(k) > 2*k), or deficiency (sigma(k) < 2*k), of a positive integer k. The measure follows from the known facts that Sum_{j=1..k} (sigma(j) + k mod j) = k^2 and that the average order of sigma(k)/k is Pi^2/6 = zeta(2) (see derivation below).

M(k) ~ 0 when sigma(k) ~ 2*k and for sufficiently large k, M(k) is positive when k is an abundant number (A005101) and negative when k is a deficient number (A005100).

The terms of this sequence are the deficient k for which M(k) < M(m) for all m < k and may be thought of as "superdeficient", contra-analogous to the superabundant numbers A004394 utilizing sigma(k)/k as the measure of abundance, which is otherwise not particularly meaningful as a deficiency measure.

15119=13*1163 is the first term that is composite and subsequently, up to 1000000000, roughly half of the terms are composite.

Links

Table of n, a(n) for n=1..36.

Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemmann Hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Amer. Math. Monthly, 109 (2002), 534-543.

Formula

Derived starting with lemmas 1-3:

1) Sum_{j=1..k} (sigma(j) + k mod j) = k^2.

2) The average order of sigma(k)/k is Pi^2/6 = zeta(2).

3) R(k) = Sum_{j=1..k} k mod j, so R(k)/k is the average order of (k mod j).

Then:

Sum_{j=1..k} sigma(j) ~ zeta(2)*Sum_{j=1..k} j = zeta(2)*(k^2+k)/2.

R(k)/k ~ k - k*zeta(2)/2 - zeta(2)/2.

0 ~ (k+1)*(1 - zeta(2)/2) - 1 - R(k)/k.

Thus M(k) = (k+1)*(1 - zeta(2)/2) - 1 - R(k)/k is a measure of variance about sigma(k) ~ 2*k corresponding to M(k) ~ 0.

Example

First few terms with their M(k) measure and factorizations as generated by the Mathematica program:
    1   -0.64493406684822643647   {{1,1}}
    5   -0.73480220054467930942   {{5,1}}
   11   -0.86960440108935861883  {{11,1}}
   23   -1.0000783673961085420   {{23,1}}
   47   -1.0528856894638174541   {{47,1}}
   59   -1.1107338698535727552   {{59,1}}
  167   -1.1984137110594038972  {{167,1}}
  179   -1.2619431113124463216  {{179,1}}
  359   -1.3499704727921791778  {{359,1}}
  503   -1.3722914063892448936  {{503,1}}
  719   -1.4363475145965658088  {{719,1}}

Mathematica

Clear[min, Rp, R, seqtable, M]; min = 1; Rp = 0; seqtable = {};
Do[R = Rp + 2 k - 1 - DivisorSigma[1, k];
  M = N[(k + 1)*(1 - Zeta[2]/2) - 1 - R/k, 20];
  If[M < min, min = M; Print[k, "   ", min, "   ", FactorInteger[k]];
   AppendTo[seqtable, k]];
  Rp = R, {k, 1, 1000000000}];
Print[seqtable]

Prog

(PARI) M(n) = (n+1)*(1 - zeta(2)/2) - 1 - sum(k=2, n, n%k)/n;
lista(nn) = my(m=+oo, list=List()); for (n=1, nn, my(mm = M(n)); if (mm < m, listput(list, n); m = mm); ); Vec(list); \\ Michel Marcus, Apr 21 2023

Crossrefs

Cf. A362081 (analogous to superabundant A004394).

Cf. A362083 (analogous to A335067, A326393).

Cf. A004490, A002201, A005100, A005101, A004125, A024916, A000290, A120444, A235796, A000396, A000079.

Sequence in context: A107010 A175942 A181669 * A306662 A052940 A191304

Adjacent sequences: A362079 A362080 A362081 * A362083 A362084 A362085

Keyword

nonn

Author

Richard Joseph Boland, Apr 17 2023

Status

approved