A002201 -id:A002201 - cp's OEIS frontend

A128701 Highly abundant numbers that are not products of consecutive primes with nonincreasing exponents, i.e., that are not of the form n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p.

1, 3, 10, 18, 20, 42, 84, 90, 108, 168, 300, 336, 504, 540, 600, 630, 660, 1008, 1200, 1560, 1620, 1980, 2100, 2340, 2400, 3024, 3120, 3240, 3780, 3960, 4200, 4680, 5880, 6120, 6240, 7920, 8400, 8820

Offset: 1

Ant King, Mar 28 2007

nonn

This is the subsequence of those highly abundant numbers (A002093) that have a different canonical structure to the superabundant numbers (A004394), the colossally abundant numbers (A004490), the highly composite numbers (A002182) and the superior highly composite numbers (A002201).

			As 10 is the third highly abundant number that cannot be expressed as a product of consecutive primes with nonincreasing exponents, then a(3)=10.

Amiram Eldar, Table of n, a(n) for n = 1..8404
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, arXiv:math/0008177 [math.NT], 2000-2001.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.
Wikipedia, Highly Abundant Numbers.

Cf. A002093, A004394, A000203, A004490, A002182, A002201, A128699, A128700, A128702.

hadata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10000}]]; data1=Flatten[Position[hadata1,#,1,1]&/@Union[hadata1]];primefactorlist[1]={1};primefactorlist[k_]:=First[Transpose[FactorInteger[k]]];exponentlist[1]={1};exponentlist[k_]:=Last[Transpose[FactorInteger[k]]];g[k_List]:=If[MemberQ[Table[k[[i]]<= k[[i-1]],{i,1,Length[k]}],False],False,True];h[k_]:=If[primefactorlist[k]==(Prime[ # ]&/@Range[Length[primefactorlist[k]]]),True,False];Select[data1,Or[ ! h[ # ],!g[exponentlist[ # ]]]&]
seq = {1}; sm = 0; Do[f = FactorInteger[n]; p = f[[;; , 1]]; e = f[[;; , 2]]; s = Times @@ ((p^(e + 1) - 1)/(p - 1)); If[s > sm, sm = s; m = Length[p]; If[p[[-1]] != Prime[m] || (m > 1 && ! AllTrue[Differences[e], # <= 0 &]), AppendTo[seq, n]]], {n, 2, 10^4}]; seq (* Amiram Eldar, Jun 18 2019 *)

The highly abundant numbers (A002093) are those values of n for which sigma(n)>sigma(m) for all mA000203(n).

A128702 Highly abundant numbers (A002093) that are not Harshad numbers (A005349).

Original entry on oeis.org

16, 96, 168, 47880, 85680, 95760, 388080, 458640, 526680, 609840, 637560, 776160, 887040, 917280, 942480, 1219680, 1244880, 1607760, 1774080, 2439360, 3880800, 5266800, 5569200, 6098400, 7761600, 9424800, 12196800, 17907120, 20900880

Offset: 1

Ant King, Mar 28 2007

All superabundant numbers (A004394), colossally abundant numbers (A004490), highly composite numbers (A002182) and superior highly composite numbers (A002201) are Harshad numbers. However, this is not true of the highly abundant numbers (A002093) and there are 32 exceptions in the 394 highly abundant numbers less than 50 million.

The previous comment is erroneous. The first superabundant number that is not a Harshad number is A004394(105) = 149602080797769600. The first highly composite number that is not a Harshad number is A002182(61) = 245044800. For all exceptions I found, the sum of digits is a power of 3. Although the first 60000 terms of the colossally abundant numbers and the superior highly composite numbers are Harshad numbers, I am not aware of a proof that all terms are Harshad numbers. There may be large counterexamples. [T. D. Noe, Oct 27 2009]

			The third highly abundant number that is not a Harshad number is 168. So a(3)=168.

Giovanni Resta, Table of n, a(n) for n = 1..400
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Wikipedia, Highly Abundant Numbers.
Wikipedia, Harshad Number.

Cf. A002093, A005349, A000203, A004394, A004490, A002182, A002201, A128699, A128700, A128701.

hadata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10^6}]]; data1=Flatten[Position[hadata1,#,1,1]&/@Union[hadata1]];HarshadQ[k_]:=If[IntegerQ[ k/(Plus @@ IntegerDigits[ k ])],True,False];Select[data1,!HarshadQ[ # ] &]

The highly abundant numbers (A002093) are those values of n for which sigma(n)>sigma(m) for all mA000203(n). Harshad numbers (A005349) are divisible by the sum of their digits.

a(16)-a(29) from Donovan Johnson, May 09 2009

A263572 Numbers n that minimize sigma(n) / (n^(1-delta) d(n)) for some delta > 0, where d = divisor count = A000005, sigma = divisor sum = A000203.

Original entry on oeis.org

1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 2248776129600, 65214507758400, 195643523275200, 6064949221531200, 12129898443062400, 448806242393308800, 18401055938125660800

Offset: 1

William C. Jagy, Oct 21 2015

nonn

These are the miserable average divisor numbers. Similar to superior highly composite numbers (A002201) or colossally abundant numbers (A004490).

a(2)..a(26) are also the first 25 terms of A002201.

William C. Jagy, Table of n, a(n) for n = 1..93
Mathematics Stack Exchange, Bounds on the average of the divisors of natural numbers

Cf. A000005, A000203, A002201, A004490.

A305056 a(n) = A004394(n)/A002110(A001221(A004394(n))).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 6, 8, 2, 4, 6, 8, 12, 24, 4, 6, 8, 12, 24, 48, 72, 120, 12, 24, 48, 72, 120, 144, 240, 288, 24, 48, 72, 120, 144, 240, 288, 360, 720, 72, 120, 144, 240, 288, 360, 720, 72, 1440, 2160, 120, 144, 240, 288, 360, 720, 1440, 2160, 2880, 4320, 5040

Offset: 1

Michael De Vlieger, Jul 01 2018

nonn

This sequence is analogous to A301413, which pertains to A002182.
Since A002182(20) = 7560 is not in A004394, a(20) =/= A301413(20), i.e., the former is 36, the latter 48. (The number 36 is not in this sequence, but is in A301413.)
A004394(50) = 120*A002110(8) is the smallest number in A004394 but not in A002182; in A004394 we have 120*A002110(m) for 4 <= m <= 8, while in A002110 we have 120*A002110(m) for 4 <= m <= 7. Therefore this sequence has one more instance of 120 (= a(50)) than exists in A301413.

			Let m be a value in this sequence. The table below shows m*A002110(A001221(A004394(k))). Columns are A001221(A004394(k)), rows are m whose products m*A002110(A001221(A004394(k))) appear in A004394 are in this sequence. Numbers in A004394 that also appear in A004490 are followed by (*).
         0     1     2      3      4       5         6  ...
      +----------------------------------------------------
   1  |  1     2*    6*
   2  |        4    12*    60*
   4  |             24    120*   840
   6  |             36    180   1260
   8  |             48    240   1680
  12  |                   360*  2520*  27720
  24  |                   720   5040*  55440*   720720*
Up to this point, the graph of this sequence and that of A301413 are identical; the asterisks pertain to numbers in A002201 in the case of A301413, but all the numbers on the graph are found in both A004490 and A002201, i.e., in A224078. The next two rows of the graph of A301413:
       0     1     2      3      4       5         6  ...
      +----------------------------------------------------
  36  |                         7560   83160   1081080
  48  |                        10080  110880   1441440*
  ...
but this sequence does not have row m = 36, as {7560, 83160, 1081080} are not in A004394.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A001221, A002110, A004394, A301413.

Block[{s = Array[DivisorSigma[1, #]/# &, 10^6], t}, t = Union@ FoldList[Max, s]; Map[#/Product[Prime@ i, {i, PrimeNu@ #}] &@ First@ FirstPosition[s, #] &, t]]

A338786 Numbers in A166981 that are neither superior highly composite nor colossally abundant.

Original entry on oeis.org

1, 4, 24, 36, 48, 180, 240, 720, 840, 1260, 1680, 10080, 15120, 25200, 27720, 110880, 166320, 277200, 332640, 554400, 665280, 2162160, 3603600, 7207200, 8648640, 10810800, 36756720, 61261200, 73513440, 122522400, 147026880, 183783600, 698377680, 735134400, 1102701600

Offset: 1

Michael De Vlieger, Nov 09 2020

These are numbers both highly composite and superabundant but neither superior highly composite nor colossally abundant.
This sequence, A224078, A304234, and A304235 are mutually exclusive subsets that comprise A166981.
Superset A166981 has 449 terms; this sequence has 358, A224078 has 20, A304234 has 39, and A304235 has 32.

			1 is in the sequence since it is the empty product, setting records for both the number of divisors and the sum of divisors, and it is neither also superior highly composite nor colossally abundant.
2 is not in the sequence since it is both colossally abundant and superior highly composite.
4 is in the sequence since it sets a record for the divisor counting and divisor sum functions, yet it is neither superior highly composite nor colossally abundant.
20951330400 is not in the sequence since it is colossally abundant though it is an HCN and SA. etc.

Michael De Vlieger, Table of n, a(n) for n = 1..358
Michael De Vlieger, Annotated plot of a(n) at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))
Eric Weisstein's World of Mathematics, Highly Composite Number
Eric Weisstein's World of Mathematics, Colossally Abundant Number
Eric Weisstein's World of Mathematics, Superabundant Number
Eric Weisstein's World of Mathematics, Superior Highly Composite Number

Cf. A189228, A002201, A002182, A004394, A004490, A166981, A224078, A304234, A304235.

Complement[Import["https://oeis.org/A166981/b166981.txt", "Data"][[1 ;; 449, -1]], Union[FoldList[Times, Import["https://oeis.org/A073751/b073751.txt", "Data"][[1 ;; 120, -1]] ], FoldList[Times, Import["https://oeis.org/A000705/b000705.txt", "Data"][[1 ;; 120, -1]] ] ] ] (* Program reads OEIS b-files Michael De Vlieger, Nov 09 2020 *)

Complement of (the union of A002182 and A004394) and (the union of A002201 and A004490).

A208767 Generalized 2-super abundant numbers.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 720, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 360360, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 10810800, 12252240, 21621600, 24504480, 36756720, 61261200

Offset: 1

Ben Branman, Mar 01 2012

nonn

The generalized k-super abundant numbers are those such that sigma_k(n)/(n^k) > sigma_k(m)/(m^k) for all m < n, where sigma_k(n) is the sum of the k-th powers of the divisors of n.
1-super abundant numbers are A004394. 0-super abundant numbers are A002182.
Pillai called these numbers "highly abundant numbers of the 2nd order". - Amiram Eldar, Jun 30 2019

			For i=24, sigma_2(24)/(24^2)=850/576=1.47569, a new record, thus 24 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..251
S. Sivasankaranarayana Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
Srinivasa Ramanujan, Highly composite numbers, Annotated and with a foreword by Jean-Louis Nicolas and Guy Robin, The Ramanujan Journal, Vol. 1, No. 2 (1997), pp. 119-153, alternative link.

Cf. A002182, A004394, A004490, A002201, A001157, A013661, A017667, A017668.

Subsequence of A025487.

s = {1}; a = 1; Do[ If[DivisorSigma[2, n]/(n^2) > a, a = DivisorSigma[2, n]/(n^2); AppendTo[s, n]], {n, 10000000}]; s

Limit_{n->oo} A001157(a(n))/a(n)^2 = zeta(2) (A013661). - Amiram Eldar, Sep 25 2022

More terms from Amiram Eldar, May 12 2019

A301415 Number of terms m in A002110 such that A301413(k) * A002110(m) is in A002182.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 4, 3, 3, 5, 3, 4, 4, 5, 5, 5, 4, 3, 4, 4, 4, 6, 3, 4, 5, 4, 3, 4, 3, 7, 5, 5, 6, 9, 6, 5, 8, 6, 8, 8, 8, 6, 6, 8, 6, 5, 7, 8, 9, 5, 5, 7, 6, 5, 6, 5, 6, 5, 6, 9, 9, 6, 9, 9, 6, 6, 7, 8, 7, 7, 7, 9, 5, 10, 10, 5, 13, 9, 9, 8, 10, 10, 7, 10, 8

Offset: 1

Michael De Vlieger, Apr 09 2018

nonn

Numbers m = A301414(x) * A002110(y) that are in A002182 are plotted below. Those also in A002201 are followed by asterisk.
This sequence counts the terms in each column.
      1     2     3     4     5     6      7 ...
  +-----------------------------------------
0 |   1
1 |   2*    4
2 |   6*   12*   24    36    48
3 |        60*  120*  180   240   360*   720
4 |             840  1260  1680  2520*  5040*
5 |                             27720  55440*
6 |                                   720720*
...

			a(1) = 3 since A301414(1) = 1 produces 3 highly composite numbers when multiplied by primorials p_0#, p_1#, and p_2# = {1, 2, 6}.
a(2) = 3 since A301414(2) = 2 yields 3 HCNs, multiplied by p_1#, p_2#, and p_3# = {4, 12, 60}.

Michael De Vlieger, Table of n, a(n) for n = 1..8000
Michael De Vlieger, Plot m = A301414(x) * A002110(y) at (x,y) for all 779674 m in Achim Flammenkamp's dataset SHCNs are shown in red; 8563 * 4096 pixels.

Cf. A002110, A002182, A301413, A301414.

f[n_] := With[{d = FactorInteger@ n}, If[n == 1, {0}, ReplacePart[Table[0, {PrimePi[d[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, d]]]; Take[#, 85] &@ Block[{s = a002182, a, b, c, m, u}, s = Take[s, 1000]; a = Array[{#2, #1, StringTrim[StringReplace[ToString@ #, ", " -> "."], ("{" | "}") ...] &[#3 /. {} -> 0], Times @@ MapIndexed[Prime[First@ #2]^#1 &, #3]} & @@ {#1, Boole[First@ #2 > 0] Length@ #2, DeleteCases[-1 + #2, 0] /. -1 -> 0} & @@ {s[[#]], f@ s[[#]]} &, Length@ s]; u = Union@ a[[All, -1]]; b = MapIndexed[{i_, j_, k_, #1} -> ToExpression@ StringJoin["{i,", ToString@ First@ #2, ",", " j, k}"] &, Union@ a[[All, -1]]]; c = Map[# /. b &, a]; m = Max[c[[All, 2]] ]; c = Map[Sort@ # &, SplitBy[SortBy[c, First], First]]; Total /@ Transpose@ Array[With[{t = ConstantArray[0, m]}, ReplacePart[t, Map[#2 -> 1 & @@ # &, c[[#]] ] ] ] &, Length@ c] ]

A309016 Superior 2-highly composite numbers: 3-smooth numbers (A003586) k for which there is a real number e > 0 such that d(k)/k^e >= d(j)/j^e for all 3-smooth numbers j, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 6, 12, 24, 72, 144, 288, 864, 1728, 5184, 10368, 20736, 62208, 124416, 373248, 746496, 1492992, 4478976, 8957952, 26873856, 53747712, 107495424, 322486272, 644972544, 1289945088, 3869835264, 7739670528, 23219011584, 46438023168, 92876046336, 278628139008, 557256278016

Offset: 1

Amiram Eldar, Jul 06 2019

nonn

How is this related to A163895? - R. J. Mathar, May 05 2023

			From _Michael De Vlieger_, Jul 12 2019: (Start)
We can plot all terms in A003586 with the power range 2^x with x >= 0 and 3^y with y >= 0 on the x and y axis, respectively. Plot of terms m in A309015, with terms also in a(n) placed in brackets:
                                2^x
          0    1     2     3     4     5     6     7     8
        +-----------------------------------------------------
     0  |[1]  [2]    4
     1  |     [6]  [12]  [24]   48
3^y  2  |           36   [72] [144]  [288]   576
     3  |                216   432   [864] [1728] 3456  6912 ...
          ...
Larger scale plot with "." representing a term m in A309015, and "o" representing a term in A309015 also in a(n) for all m < A002110(20).
                              2^x
        0    5   10   15   20   25   30   35   40   45  ...
        +------------------------------------------------
       0|oo.
        | ooo.
        |  .ooo.
        |   ..oo..
        |    ..ooo..
       5|      ..oo...
        |       ..ooo...
        |         ..oo....
        |          ..ooo....
        |            ..ooo....
      10|             ...oo.....
        |               ..ooo....
        |                ...oo.....
        |                  ..ooo.....
3^y     |                   ...ooo....
      15|                     ...oo.....
        |                      ...ooo.....
        |                        ...oo.....
        |                         ...ooo.....
        |                           ...oo......
      20|                            ...ooo.....
        |                              ...ooo.....
        |                               ....oo......
        |                                 ...ooo.....
        |                                  ....oo......
      25|                                    ...ooo......
        |                                     ....ooo....
        |                                       ....oo.
        |                                        ....o
        |                                          .
     ...
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..2709
Gérard Bessi, Etude des nombres 2-hautement composés, Séminaire de Théorie des nombres de Bordeaux, Vol. 4 (1975), pp. 1-22.
Michael De Vlieger, Factors p analogous to A000705 such that the product of the smallest n terms equals a(n + 1) (10^5 terms).

Subsequence of A003586 and A309015.

Cf. A000005, A002201.

f[nn_, k_: 2] := Block[{w = {{2, 1}, {3, 0}}, s = {2}, P = 1, q = k - 2, x, i, n, f}, f[w_List] := Log[#1, (#2 + 2)/(#2 + 1)] & @@ w; x = Array[f[w[[#]] ] &, P + 1]; For[n = 2, n <= nn, n++, i = First@ FirstPosition[x, Max[x]]; AppendTo[s, w[[i, 1]]]; w[[i, 2]]++; If[And[i > P, P <= q], P++; AppendTo[w, {Prime[i + 1], 0}]; AppendTo[x, f[Last@ w]]]; x[[i]] = f@ w[[i]] ]; s]; {1}~Join~FoldList[Times, f[32, 2]] (* Michael De Vlieger, Jul 11 2019, after T. D. Noe at A000705 *)

More terms from Michael De Vlieger, Jul 11 2019

A340840 Union of the highly composite and superabundant numbers.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880

Offset: 1

Michael De Vlieger, Jan 27 2021

nonn

Numbers m that set records in A000005 and numbers k that set records for the ratio A000203(k)/k, sorted, with duplicates removed.
All terms are in A025487, since all terms in A002182 and A004394 are products of primorials P in A002110.
For numbers that are highly composite but not superabundant, see A308913; for numbers that are superabundant but not highly composite, see A166735. - Jon E. Schoenfield, Jun 14 2021

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated scatterplot of a(n) = A301414(x) * A002110(y) at (x, y) showing the intersection A166981 of A002182 and A004394, and the intersection A224078 of A002201 and A004490.

Cf. A000005, A000203, A027750, A002110.
Supersets: A025487, A055932.
Subsets: A002182, A002201, A004394, A004490, A166735, A166981, A168263, A189228, A224078, A304234, A304235, A308913, A333655, A338786.

(* Load the function f[] at A025487, then: *) Block[{t = Union@ Flatten@ f[15], a = {}, b = {}, d = 0, s = 0}, Do[(If[#2 > d, d = #2; AppendTo[a, #1]]; If[#3/#1 > s, s = #3/#1; AppendTo[b, #1]]) & @@ Flatten@ {t[[i]], DivisorSigma[{0, 1}, t[[i]]]}, {i, Length@ t}]; Union[a, b]]

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.

cp's OEIS Frontend

A128701 Highly abundant numbers that are not products of consecutive primes with nonincreasing exponents, i.e., that are not of the form n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p.

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A128702 Highly abundant numbers (A002093) that are not Harshad numbers (A005349).

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A263572 Numbers n that minimize sigma(n) / (n^(1-delta) d(n)) for some delta > 0, where d = divisor count = A000005, sigma = divisor sum = A000203.

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A305056 a(n) = A004394(n)/A002110(A001221(A004394(n))).

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A338786 Numbers in A166981 that are neither superior highly composite nor colossally abundant.

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A208767 Generalized 2-super abundant numbers.

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A301415 Number of terms m in A002110 such that A301413(k) * A002110(m) is in A002182.

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A309016 Superior 2-highly composite numbers: 3-smooth numbers (A003586) k for which there is a real number e > 0 such that d(k)/k^e >= d(j)/j^e for all 3-smooth numbers j, where d(k) is the number of divisors of k (A000005).

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