A326393 -id:A326393 - cp's OEIS frontend

A335067 Numbers k where records occur for sigma(k+1)/sigma(k), where sigma(k) is the sum of divisors of k (A000203).

1, 179, 239, 359, 719, 839, 1259, 3359, 5039, 10079, 25199, 27719, 50399, 55439, 110879, 166319, 360359, 665279, 831599, 1081079, 1441439, 2162159, 3603599, 4324319, 12972959, 21621599, 43243199, 61261199, 73513439, 122522399, 205405199, 245044799, 410810399

Offset: 1

Amiram Eldar, May 22 2020

nonn

Shapiro (1978) proved that the closure of the set {sigma(k+1)/sigma(k) | k >= 1} consists of all the nonnegative reals. In particular, sigma(k+1)/sigma(k) is unbounded and therefore this sequence is infinite.

25199 is the first composite term.

			The values of sigma(k+1)/sigma(k) for the first terms are 3, 3.033..., 3.1, 3.25, 3.358..., ...

Roy E. DeMeo, Jr., Problem 6107, Advanced Problems, The American Mathematical Monthly, Vol. 83, No. 7 (1976), p. 573, The Closure of sigma(n+1)/sigma(n), solution by Harold N. Shapiro, ibid., Vol. 85, No. 4 (1978), pp. 287-289.

Cf. A000203, A282531, A326393, A335068.

rm = 0; s1 = 1; seq = {}; Do[s2 = DivisorSigma[1,n]; If[(r = s2/s1) > rm, rm = r; AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq

A362081 Numbers k achieving record abundance (sigma(k) > 2*k) via a residue-based measure M(k) (see Comments), analogous to superabundant numbers A004394.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 36, 72, 120, 360, 420, 840, 1680, 2520, 4032, 5040, 10080, 25200, 32760, 65520, 98280, 194040, 196560, 388080, 942480, 1801800, 3160080, 3603600, 6320160, 12640320, 24504480, 53721360, 61981920, 73513440, 115315200, 122522400, 189909720, 192099600, 214885440

Offset: 1

Richard Joseph Boland, Apr 08 2023

nonn

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).

			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}}

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. A362082 (superdeficient), A362083 (analogous to A335067, A326393).

Cf. A004394, A004490, A002201, A005100, A005101, A004125, A024916, A000290, A120444, A235796, A000396, A000079.

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]

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.

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.

Original entry on oeis.org

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.

A335068 Numbers k where records occur for sigma(k)/sigma(k+1), where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 240, 420, 840, 1680, 2520, 5040, 7560, 12600, 15120, 30240, 55440, 110880, 221760, 332640, 665280, 720720, 1441440, 2882880, 3603600, 4324320, 10810800, 24504480, 36756720, 41081040, 43243200, 64864800, 73513440, 122522400, 183783600

Offset: 1

Amiram Eldar, May 22 2020

nonn

Shapiro (1978) proved that the closure of the set {sigma(k+1)/sigma(k) | k >= 1} consists of all the nonnegative reals. In particular, sigma(k+1)/sigma(k) can be arbitrarily close to 0 and thus sigma(k)/sigma(k+1) is unbounded and this sequence is infinite.

			The values of sigma(k)/sigma(k+1) for the first terms are 0.333..., 0.75, 1.166..., 1.5, 2, ...

Roy E. DeMeo, Jr., Problem 6107, Advanced Problems, The American Mathematical Monthly, Vol. 83, No. 7 (1976), p. 573, The Closure of sigma(n+1)/sigma(n), solution by Harold N. Shapiro, ibid., Vol. 85, No. 4 (1978), pp. 287-289.

Cf. A000203, A282531, A326393, A335067.

rm = 0; s1 = 1; seq = {}; Do[s2 = DivisorSigma[1,n]; If[(r = s1/s2) > rm, rm = r; AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq
With[{nn=721000},DeleteDuplicates[Thread[{Range[nn-1],#[[1]]/#[[2]]&/@Partition[ DivisorSigma[ 1,Range[nn]],2,1]}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* The program generates the first 25 terms of the sequence. *) (* Harvey P. Dale, Jan 12 2024 *)

cp's OEIS Frontend

A335067 Numbers k where records occur for sigma(k+1)/sigma(k), where sigma(k) is the sum of divisors of k (A000203).

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A362081 Numbers k achieving record abundance (sigma(k) > 2*k) via a residue-based measure M(k) (see Comments), analogous to superabundant numbers A004394.

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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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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.

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A335068 Numbers k where records occur for sigma(k)/sigma(k+1), where sigma(k) is the sum of divisors of k (A000203).

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Mathematica