A004394 -id:A004394 - cp's OEIS frontend

A166981 Superabundant numbers (A004394) that are highly composite (A002182).

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800

Offset: 1

T. D. Noe, Oct 26 2009

The intersection of superabundant and highly composite numbers has exactly 449 terms, the largest of which is 2^10 * 3^6 * 5^4 * 7^3 * 11^3 * 13^2 * 17^2 * 19^2 * 23^2 * 29 * 31 * 37*...*347.
The argument showing that this is a finite sequence seems to be given in A166735. - N. J. A. Sloane, Jan 04 2019
Pillai proved that this sequence is finite and asked for its number of terms (he used the term "highly abundant" for superabundant numbers). - Amiram Eldar, Jun 30 2019
From Michael De Vlieger, Dec 29 2020: (Start)
All terms are products of primorials (numbers in A002110), thus, all terms are also in A025487, itself a subsequence of A055932.
Since the colossally abundant numbers (CA, A004490) are also superabundant, and since the superior highly composite (SHC A002201) numbers are also highly composite, the finite sequence A224078 containing numbers both CA and SHC is a subsequence of this sequence. Likewise, A304234 (numbers that are SA, HC, & SHC but not CA) and A304235 (numbers that are SA, HC, & CA but not SHC), and A338786 (SA and HC, but neither CA nor SHC) are mutually exclusive finite subsequences of this sequence. (End)

T. D. Noe, Table of n, a(n) for n = 1..449 (complete sequence)
Michael De Vlieger, Annotated plot of a(n) at (x,y) = (a(n)/P), P) where P = A002110(A001221(a(n)) showing all 449 terms.
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. Mentions this sequence.
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.

Cf. A002110, A002182, A004394, A025487, A055932, A166735 (SA numbers that are not HC numbers), A224078, A304234, A304235, A308913, A338786.

A189228 Superabundant numbers (A004394) that are not colossally abundant (A004490).

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

Offset: 1

Jonathan Sondow, Jun 07 2011

nonn

All colossally abundant (CA) numbers are also superabundant (SA). (Proof. If n is CA and k < n, then sigma(n)/n = n^{epsilon}*sigma(n)/n^{1+epsilon} >= n^{epsilon}*sigma(k)/k^{1+epsilon} > k^{epsilon}*sigma(k)/k^{1+epsilon} = sigma(k)/k, and so n is SA.)

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A112974 (Number of superabundant numbers between two consecutive colossally abundant numbers) and A166735 (Superabundant numbers that are not highly composite).

A166735 Superabundant numbers (A004394) that are not highly composite (A002182).

Original entry on oeis.org

1163962800, 4658179125600, 13974537376800, 144403552893600, 433210658680800, 10685862914126400, 21371725828252800, 32057588742379200, 37400520199442400, 64115177484758400, 1533421328177138400

Offset: 1

T. D. Noe, Oct 20 2009

nonn

Alaoglu and Erdos mention the first term in footnote 14.

Because the "shapes" of superabundant and highly composite numbers are different, there is a last superabundant number that is also highly composite. In factored form, that 154-digit number is N = A004394(1023) = A002182(2567) = 2^10 3^6 5^4 7^3 11^3 13^2 17^2 19^2 23^2 29 31 37...347. In other words, this sequence contains all superabundant numbers greater than N. - T. D. Noe, Oct 26 2009

T. D. Noe, Table of n, a(n) for n = 1..574
L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. Mentions this sequence.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.

Cf. A166981 (intersection of SA and HC numbers). - T. D. Noe, Oct 26 2009

Cf. A189228 (SA numbers that are not CA).

a(574+i) = A004394(1023+i) for i>0.

A007626 Sum of divisors of superabundant numbers (A004394).

Original entry on oeis.org

1, 3, 7, 12, 28, 60, 91, 124, 168, 360, 546, 744, 1170, 2418, 2880, 4368, 5952, 9360, 19344, 39312, 59520, 99944, 112320, 232128, 471744, 714240, 1199328, 1451520, 2437344, 2926080, 3249792, 6604416, 9999360

Offset: 1

Walter Nissen

Local maxima of sigma(n), the sum of divisors function A000203.

Same as A063072 for the first 19 terms. - T. D. Noe, Jul 01 2008

Amiram Eldar, Table of n, a(n) for n = 1..6863 (terms 1..500 from T. D. Noe)

Cf. A002192, A000203, A051444, A004394.

See A034885 and A002093 for another version.

Reap[ For[ n=1; a=0, n <= 3*10^6, n++, s = DivisorSigma[1, n]; b = s/n; If[ b>a, a=b; Print[s]; Sow[s]]]][[2, 1]] (* Jean-François Alcover, Apr 02 2013 *)
Join[{1},DeleteDuplicates[Select[{#[[1]],#[[2]],#[[2]]/#[[1]]}&/@Table[ {n,DivisorSigma[1,n]}, {n,10^6}],#[[3]]>1&],GreaterEqual[#1[[3]],#2[[3]]]&][[All,2]]] (* The program generates the first 31 terms of the sequence. *) (* Harvey P. Dale, Oct 04 2022 *)

a(n) = A000203(A004394(n)). - Amiram Eldar, Sep 25 2021

A189686 Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

Original entry on oeis.org

2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 2520, 5040

Offset: 1

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, May 30 2011

nonn

5040 is the last element in the sequence if and only if the Riemann Hypothesis is true. (See Akbary and Friggstad in A004394.)

G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33 (see Table 1).
G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.

Cf. A004394, A091901, A067698, A166981, A077006.

kmax = 10^4;
A004394 = Join[{1}, Reap[For[r = 1; k = 2, k <= kmax, k = k + 2, s = DivisorSigma[-1, k]; If[s > r, r = s; Sow[k]]]][[2, 1]]];
A067698 = Select[Range[2, kmax], DivisorSigma[1, #] > Exp[EulerGamma] # Log[Log[#]]&];
Intersection[A004394, A067698] (* Jean-François Alcover, Jan 28 2019 *)

is(n)=sigma(n) >= exp(Euler) * n * log(log(n)); \\ A067698
lista(nn) = my(r=1, t); forstep(n=2, nn, 2, t=sigma(n, -1); if(t>r && is(n), r=t; print1(n, ", "))); \\ Michel Marcus, Jan 28 2019; adapted from A004394

Equals A004394 intersect A067698.

Erroneous terms 1260 and 1680 removed by Jean-François Alcover, Jan 28 2019

A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

Original entry on oeis.org

3, 8, 10, 16, 18, 20, 30, 42, 72, 84, 90, 96, 108, 144, 168, 210, 216, 288, 300, 336, 420, 480, 504, 540, 600, 630, 660, 960, 1008, 1080, 1200, 1440, 1560, 1620, 1800, 1920, 1980, 2100, 2160, 2340, 2400, 2880, 3024, 3120, 3240, 3360, 3600, 3780, 3960, 4200

Offset: 1

Ant King, Mar 28 2007

nonn

In 1944, Alaoglu and Erdős conjectured that this sequence was infinite and this was proved to be true by Nicolas in 1969.

			The sequence of highly abundant numbers begins 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20 and the sequence of superabundant numbers begins 1, 2, 4, 6, 12, 24. Because 10 is the third number which is in the first sequence but not in the second, it follows that a(3)=10.

Amiram Eldar, Table of n, a(n) for n = 1..1000
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers" Bull. Soc. Math. France 97 (1969), pp. 129-191.
Eric Weisstein's World of Mathematics, Superabundant Number.
Wikipedia, Highly Abundant Numbers.

Cf. A000203, A002093, A004394, A128700, A128701, A128702.

habdata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10000}]]; data1=Flatten[Position[habdata1,#,1,1]&/@Union[habdata1]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; data2=Flatten[Position[sabdata2,#,1,1]&/@Union[sabdata2]];sabdata2=FoldList[Max,1,Table[DivisorSigma[1,n]/n,{n,2,10000}]]; Complement[data1,data2]

The highly abundant numbers are those integers for which sigma(n) > sigma(m) for all m < n (A002093) and the superabundant numbers are those integers for which sigma(n)/n > sigma(m)/m for all m < n (A004394).

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]]

A308913 Highly composite numbers (A002182) that are not superabundant numbers (A004394).

Original entry on oeis.org

7560, 20160, 45360, 50400, 83160, 221760, 498960, 1081080, 2882880, 6486480, 14414400, 17297280, 32432400, 43243200, 110270160, 245044800, 294053760, 551350800, 2095133040, 2205403200, 4655851200, 5587021440, 10475665200, 64250746560, 73329656400, 97772875200

Offset: 1

Amiram Eldar, Jun 30 2019

nonn

Pillai noted in 1941 that 7560 is the first term of this sequence. He also asked for the opposite sequence and wondered whether its first term (A166735(1) = 1163962800) is within the reach of modern computation.

Since the sequence of superabundant numbers that are also highly composite, A166981, is finite, this sequence contains all the highly composite numbers above A002182(2567) = A004394(1023).

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.
S. Sivasankaranarayana Pillai, Highly Composite Numbers of the t th Order, J. Indian Math. Soc., Vol. 8 (1944), pp. 61-74.

Cf. A002182, A004394, A166735, A166981, A181309.

seq = {}; dm = 0; sm = 0; Do[d = DivisorSigma[0, n]; s = DivisorSigma[1, n]; If[d > dm, dm = d]; If[s > s, sm = s, AppendTo[seq, n]], {n, 1, 3000000}]; seq

a(2118+i) = A002182(2567+i) for i > 0.

A353300 Numbers k such that A004394(k)-1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 19, 20, 24, 25, 26, 30, 32, 41, 47, 48, 49, 51, 57, 59, 76, 82, 83, 92, 104, 105, 117, 119, 131, 134, 137, 139, 143, 154, 166, 170, 172, 180, 209, 214, 215, 216, 217, 227, 231, 234, 247, 265, 269, 271, 284, 317, 327, 348

Offset: 1

Amiram Eldar, Apr 10 2022

nonn

First differs from A306588 at n=15.

			3 is a term since A004394(3)-1 = 4-1 = 3 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..314

Cf. A004394, A072826, A306588, A353301, A353302.

s = {}; abm = 0; k = 0; Do[ab = DivisorSigma[-1, n]; If[ab > abm, abm = ab; k++; If[PrimeQ[n - 1], AppendTo[s, k]]], {n, 1, 10^6}]; s

A353301 Numbers k such that A004394(k)+1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 18, 21, 24, 25, 28, 35, 45, 46, 50, 56, 70, 73, 76, 78, 79, 82, 89, 94, 105, 113, 116, 118, 121, 123, 124, 138, 139, 153, 157, 159, 164, 197, 201, 203, 210, 217, 253, 261, 273, 280, 283, 287, 342, 352, 356, 381, 396, 437, 450, 471

Offset: 1

Amiram Eldar, Apr 10 2022

nonn

First differs from A306587 at n=11.

			1 is a term since A004394(1)+1 = 1+1 = 2 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A004394, A072828, A306587, A353300, A353302.

s = {}; abm = 0; k = 0; Do[ab = DivisorSigma[-1, n]; If[ab > abm, abm = ab; k++; If[PrimeQ[n + 1], AppendTo[s, k]]], {n, 1, 10^6}]; s

cp's OEIS Frontend

A166981 Superabundant numbers (A004394) that are highly composite (A002182).

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A189228 Superabundant numbers (A004394) that are not colossally abundant (A004490).

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A166735 Superabundant numbers (A004394) that are not highly composite (A002182).

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A007626 Sum of divisors of superabundant numbers (A004394).

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A189686 Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

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A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

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

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A308913 Highly composite numbers (A002182) that are not superabundant numbers (A004394).

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A353300 Numbers k such that A004394(k)-1 is prime.

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