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.
%I A362083 #28 Jun 16 2023 13:50:03
%S A362083 11,17,19,47,53,103,347,349,557,1663,1679,2519,5039,10079,15119,25199,
%T A362083 27719,55439,110879,166319,277199,332639,554399,665279,720719,1441439,
%U A362083 2162159,3603599,4324319,7207199,8648639,10810799,21621599,36756719,61261199,73513439,122522399,147026879
%N A362083 Numbers k such that, via a residue based measure M(k) (see Comments), k is deficient, k+1 is abundant, and abs(M(k)) + abs(M(k+1)) reaches a new maximum.
%C A362083 The residue-based quantifier function, M(k), measures either abundance (sigma(k) > 2*k), or deficiency (sigma(k) < 2*k), of a positive integer k. The measure is defined by M(k) = (k+1)*(1 - zeta(2)/2) - 1 - (Sum_{j=1..k} k mod j)/k. It 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 A362083 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 such that k+1 is abundant and abs(M(k)) + abs(M(k+1)) achieves a new maximum, somewhat analogous to A335067 and A326393.
%H A362083 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 A362083 Derived starting with lemmas 1-3:
%F A362083 1) Sum_{j=1..k} (sigma(j) + k mod j) = k^2.
%F A362083 2) The average order of sigma(k)/k is Pi^2/6 = zeta(2).
%F A362083 3) R(k) = Sum_{j=1..k} k mod j, so R(k)/k is the average order of (k mod j).
%F A362083 Then:
%F A362083 Sum_{j=1..k} sigma(j) ~ zeta(2)*Sum_{j=1..k} j = zeta(2)*(k^2+k)/2.
%F A362083 R(k)/k ~ k - k*zeta(2)/2 - zeta(2)/2.
%F A362083 0 ~ (k+1)*(1 - zeta(2)/2) - 1 - R(k)/k.
%F A362083 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 A362083 The first few terms with measure sums and factorizations generated by the Mathematica program:
%e A362083 0.90610439514731535319 35 {{5,1},{7,1}} 36 {{2,2},{3,2}}
%e A362083 1.1735781643159997761 59 {{59,1}} 60 {{2,2},{3,1},{5,1}}
%e A362083 1.3642976724582397229 119 {{7,1},{17,1}} 120 {{2,3},{3,1},{5,1}}
%e A362083 1.3954100615479538209 179 {{179,1}} 180 {{2,2},{3,2},{5,1}}
%e A362083 1.4600817810807682323 239 {{239,1}} 240 {{2,4},{3,1},{5,1}}
%e A362083 1.6088158511317518390 359 {{359,1}} 360 {{2,3},{3,2},{5,1}}
%e A362083 1.7153941935887132383 719 {{719,1}} 720 {{2,4},{3,2},{5,1}}
%e A362083 1.7851979872921589879 839 {{839,1}} 840 {{2,3},{3,1},{5,1},{7,1}}
%t A362083 Clear[max, Rp, R, seqtable, Mp, M];max = -1; Rp = 0; Mp = -0.644934066; seqtable = {};
%t A362083 Do[R = Rp + 2 k - 1 - DivisorSigma[1, k];
%t A362083 M = N[(k)*(1 - Zeta[2]/2) - 1 - R/k, 20];
%t A362083 If[DivisorSigma[1, k - 1] < 2 (k - 1) && DivisorSigma[1, k] > 2 k &&
%t A362083 Abs[Mp] + Abs[M] > max, max = Abs[Mp] + Abs[M];
%t A362083 Print[max, " ", k - 1, " ", FactorInteger[k - 1], " ", k,
%t A362083 " ", FactorInteger[k]]; AppendTo[seqtable, {k - 1, k}]]; Rp = R;
%t A362083 Mp = M, {k, 2, 1000000000}]; seq = Flatten[seqtable]; Table[seq[[2 j - 1]], {j, 1, Length[seq]/2}]
%Y A362083 Cf. A362081 (analogous to superabundant A004394), A362082 (superdeficient).
%Y A362083 Cf. A335067, A326393, A004490, A002201, A326393, A005100, A005101, A004125, A024916, A000290, A120444, A235796, A000396, A000079.
%K A362083 nonn
%O A362083 1,1
%A A362083 _Richard Joseph Boland_, Apr 17 2023