OFFSET
1,2
COMMENTS
The residue-based quantifier function, M(k) = (k+1)*(1 - zeta(2)/2) - 1 - ( Sum_{j=1..k} k mod j )/k, measures either abundance (sigma(k) > 2*k), or deficiency (sigma(k) < 2*k), of a positive integer 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 abundant k for which M(k) > M(m) for all m < k, analogous to the superabundant numbers A004394, which utilize sigma(k)/k as the measure. However, sigma(k)/k does not give a meaningful measure of deficiency, whereas M(k) does, thus a sensible notion of superdeficient (see A362082).
LINKS
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
The abundance measure is initially negative, becoming positive for k > 30. Initial measures with factorizations from the Mathematica program:
1 -0.64493406684822643647 {{1,1}}
2 -0.46740110027233965471 {{2,1}}
4 -0.36233516712056609118 {{2,2}}
6 -0.25726923396879252765 {{2,1},{3,1}}
12 -0.10873810118013850374 {{2,2},{3,1}}
24 -0.10334250226949712257 {{2,3},{3,1}}
30 -0.096478036147509765322 {{2,1},{3,1},{5,1}}
36 0.068719763307810925260 {{2,2},{3,2}}
72 0.12657322670640173542 {{2,3},{3,2}}
MATHEMATICA
Clear[max, Rp, R, seqtable, M];
max = -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 > max, max = M; Print[k, " ", max, " ", 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
KEYWORD
nonn
AUTHOR
Richard Joseph Boland, Apr 08 2023
STATUS
approved