A362081 -id:A362081 - cp's OEIS frontend

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

Richard Joseph Boland, Apr 17 2023

nonn

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.

			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}}

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.

Cf. A362081 (analogous to superabundant A004394).
Cf. A362083 (analogous to A335067, A326393).
Cf. A004490, A002201, A005100, A005101, A004125, A024916, A000290, A120444, A235796, A000396, A000079.

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]

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

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.

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.

Original entry on oeis.org

11, 17, 19, 47, 53, 103, 347, 349, 557, 1663, 1679, 2519, 5039, 10079, 15119, 25199, 27719, 55439, 110879, 166319, 277199, 332639, 554399, 665279, 720719, 1441439, 2162159, 3603599, 4324319, 7207199, 8648639, 10810799, 21621599, 36756719, 61261199, 73513439, 122522399, 147026879

Offset: 1

Richard Joseph Boland, Apr 17 2023

nonn

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).

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.

			The first few terms with measure sums and factorizations generated by the Mathematica program:
0.90610439514731535319   35  {{5,1},{7,1}}   36   {{2,2},{3,2}}
1.1735781643159997761    59  {{59,1}}        60   {{2,2},{3,1},{5,1}}
1.3642976724582397229   119  {{7,1},{17,1}} 120   {{2,3},{3,1},{5,1}}
1.3954100615479538209   179  {{179,1}}      180   {{2,2},{3,2},{5,1}}
1.4600817810807682323   239  {{239,1}}      240   {{2,4},{3,1},{5,1}}
1.6088158511317518390   359  {{359,1}}      360   {{2,3},{3,2},{5,1}}
1.7153941935887132383   719  {{719,1}}      720   {{2,4},{3,2},{5,1}}
1.7851979872921589879   839  {{839,1}}      840   {{2,3},{3,1},{5,1},{7,1}}

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.

Cf. A362081 (analogous to superabundant A004394), A362082 (superdeficient).

Cf. A335067, A326393, A004490, A002201, A326393, A005100, A005101, A004125, A024916, A000290, A120444, A235796, A000396, A000079.

Clear[max, Rp, R, seqtable, Mp, M];max = -1; Rp = 0; Mp = -0.644934066; seqtable = {};
Do[R = Rp + 2 k - 1 - DivisorSigma[1, k];
 M = N[(k)*(1 - Zeta[2]/2) - 1  - R/k, 20];
 If[DivisorSigma[1, k - 1] < 2 (k - 1) && DivisorSigma[1, k] > 2 k &&
   Abs[Mp] + Abs[M] > max, max = Abs[Mp] + Abs[M];
  Print[max, "   ", k - 1, "   ", FactorInteger[k - 1], "   ", k,
   "   ", FactorInteger[k]]; AppendTo[seqtable, {k - 1, k}]]; Rp = R;
 Mp = M, {k, 2, 1000000000}]; seq = Flatten[seqtable]; Table[seq[[2 j - 1]], {j, 1, Length[seq]/2}]

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.

cp's OEIS Frontend

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.

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