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
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
KEYWORD
nonn
AUTHOR
Richard Joseph Boland, Apr 17 2023
STATUS
approved