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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.

%I A362082 #48 Jun 16 2023 13:48:18
%S A362082 1,5,11,23,47,59,167,179,359,503,719,1439,5039,6719,7559,15119,20159,
%T A362082 52919,75599,83159,166319,415799,720719,831599,1081079,2162159,
%U A362082 4324319,5266799,7900199,10533599,18345599,28274399,41081039,136936799,205405199,410810399
%N 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.
%C A362082 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).
%C A362082 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).
%C A362082 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.
%C A362082 15119=13*1163 is the first term that is composite and subsequently, up to 1000000000, roughly half of the terms are composite.
%H A362082 Jeffrey C. Lagarias, <a href="https://arxiv.org/abs/math/0008177">An Elementary Problem Equivalent to the Riemmann Hypothesis</a>, arXiv:math/0008177 [math.NT], 2000-2001; Amer. Math. Monthly, 109 (2002), 534-543.
%F A362082 Derived starting with lemmas 1-3:
%F A362082 1) Sum_{j=1..k} (sigma(j) + k mod j) = k^2.
%F A362082 2) The average order of sigma(k)/k is Pi^2/6 = zeta(2).
%F A362082 3) R(k) = Sum_{j=1..k} k mod j, so R(k)/k is the average order of (k mod j).
%F A362082 Then:
%F A362082 Sum_{j=1..k} sigma(j) ~ zeta(2)*Sum_{j=1..k} j = zeta(2)*(k^2+k)/2.
%F A362082 R(k)/k ~ k - k*zeta(2)/2 - zeta(2)/2.
%F A362082 0 ~ (k+1)*(1 - zeta(2)/2) - 1 - R(k)/k.
%F A362082 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.
%e A362082 First few terms with their M(k) measure and factorizations as generated by the Mathematica program:
%e A362082     1   -0.64493406684822643647   {{1,1}}
%e A362082     5   -0.73480220054467930942   {{5,1}}
%e A362082    11   -0.86960440108935861883  {{11,1}}
%e A362082    23   -1.0000783673961085420   {{23,1}}
%e A362082    47   -1.0528856894638174541   {{47,1}}
%e A362082    59   -1.1107338698535727552   {{59,1}}
%e A362082   167   -1.1984137110594038972  {{167,1}}
%e A362082   179   -1.2619431113124463216  {{179,1}}
%e A362082   359   -1.3499704727921791778  {{359,1}}
%e A362082   503   -1.3722914063892448936  {{503,1}}
%e A362082   719   -1.4363475145965658088  {{719,1}}
%t A362082 Clear[min, Rp, R, seqtable, M]; min = 1; Rp = 0; seqtable = {};
%t A362082 Do[R = Rp + 2 k - 1 - DivisorSigma[1, k];
%t A362082   M = N[(k + 1)*(1 - Zeta[2]/2) - 1 - R/k, 20];
%t A362082   If[M < min, min = M; Print[k, "   ", min, "   ", FactorInteger[k]];
%t A362082    AppendTo[seqtable, k]];
%t A362082   Rp = R, {k, 1, 1000000000}];
%t A362082 Print[seqtable]
%o A362082 (PARI) M(n) = (n+1)*(1 - zeta(2)/2) - 1 - sum(k=2, n, n%k)/n;
%o A362082 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
%Y A362082 Cf. A362081 (analogous to superabundant A004394).
%Y A362082 Cf. A362083 (analogous to A335067, A326393).
%Y A362082 Cf. A004490, A002201, A005100, A005101, A004125, A024916, A000290, A120444, A235796, A000396, A000079.
%K A362082 nonn
%O A362082 1,2
%A A362082 _Richard Joseph Boland_, Apr 17 2023