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A362083
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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.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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}}
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A335067, A326393, A004490, A002201, A326393, A005100, A005101, A004125, A024916, A000290, A120444, A235796, A000396, A000079.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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