A304235 -id:A304235 - 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.

A304234 Superior highly composite numbers that are superabundant but not colossally abundant.

Original entry on oeis.org

13967553600, 2248776129600, 65214507758400, 195643523275200, 12129898443062400, 448806242393308800, 18401055938125660800, 185942670254759802384000, 9854961523502269526352000, 1162885459773267804109536000, 780296143507862696557498656000

Offset: 1

Michael De Vlieger, May 08 2018

Numbers m in A002201 that are also in A004394 but not A004490.
Subset of A166981. Numbers in this sequence are in neither A224078 nor A304235.
There are 39 terms in this sequence.
The smallest term is 2^5 * 3^2 * 5 * A002110(8) or the product of A002110(k) with k = {1,1,1,2,3,8}.
The largest is 2^10 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 * 19 * 23 * A002110(65) or the product of A002110(k) with k = {1,1,1,1,2,2,2,3,4,9,65}, a 144 digit decimal number.

Michael De Vlieger, Table of n, a(n) for n = 1..39
Michael De Vlieger, Superior highly composite numbers m that are also superabundant but not colossally abundant
Michael De Vlieger, Color coded plot of m A002182 and A004394 at (x,y) where A301414(x) * A002110(y) = m, terms in this sequence are colored dark blue.
Michael De Vlieger, Annotated plot of a(n) for 1 <= n <= 39 at (x,y) = (a(n)/A002110(A001221(a(n)), A002110(A001221(a(n)))

Cf. A002110, A002182, A002201, A004394, A004490, A166981, A224078, A304235.

(* First, download b-files at A002201, A004394, and A004490 *)
f[w_] := Times @@ Flatten@ {Complement[#1, Union[#2, #3]], Product[Prime@ i, {i, PrimePi@ #}] & /@ #2, Factorial /@ #3} & @@ ToExpression@ {StringSplit[w, _?(! DigitQ@ # &)], StringCases[w, (x : DigitCharacter ..) ~~ "#" :> x], StringCases[w, (x : DigitCharacter ..) ~~ "!" :> x]};
With[{s = Import["b002201.txt", "Data"][[All, -1]], t = Select[Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ], IntegerQ@ First@ # &][[All, -1]], u = Import["b004490.txt", "Data"][[All, -1]]}, Select[Intersection[s, t], FreeQ[u, #] &]]

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).

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]]

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

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A304234 Superior highly composite numbers that are superabundant but not colossally abundant.

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

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A340840 Union of the highly composite and superabundant numbers.

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