A189228 -id:A189228 - cp's OEIS frontend

A004394 Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.

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, 21621600

Offset: 1

Matthew Conroy

Matthew Conroy points out that these are different from the highly composite numbers - see A002182. Jul 10 1996
With respect to the comment above, neither sequence is subsequence of the other. - Ivan N. Ianakiev, Feb 11 2020
Also n such that sigma_{-1}(n) > sigma_{-1}(m) for all m < n, where sigma_{-1}(n) is the sum of the reciprocals of the divisors of n. - Matthew Vandermast, Jun 09 2004
Ramanujan (1997, Section 59; written in 1915) called these numbers "generalized highly composite." Alaoglu and Erdős (1944) changed the terminology to "superabundant." - Jonathan Sondow, Jul 11 2011
Alaoglu and Erdős show that: (1) n is superabundant => n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2 >= e_3 >= ... >= e_p (and e_p is 1 unless n=4 or n=36); (2) if q < r are primes, then | e_r - floor(e_q*log(q)/log(r)) | <= 1; (3) q^{e_q} < 2^{e_2+2} for primes q, 2 < q <= p. - Keith Briggs, Apr 26 2005
It follows from Alaoglu and Erdős finding 1 (above) that, for n > 7, a(n) is a Zumkeller Number (A083207); for details, see Proposition 9 and Corollary 5 at Rao/Peng link (below). - Ivan N. Ianakiev, Feb 11 2020
See A166735 for superabundant numbers that are not highly composite, and A189228 for superabundant numbers that are not colossally abundant.
Pillai called these numbers "highly abundant numbers of the 1st order". - Amiram Eldar, Jun 30 2019

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.
J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad. Publ., 2002 (see pp. 19-21).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

D. Kilminster, Table of n, a(n) for n = 1..2000 (extends to n = 8436 in the comments; first 500 terms from T. D. Noe)
A. Akbary and Z. Friggstad, Superabundant numbers and the Riemann hypothesis, Amer. Math. Monthly, 116 (2009), 273-275.
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata.
Christian Axler, Inequalities involving the primorial counting function, arXiv:2406.04018 [math.NT], 2024. See p. 12.
Yu. Bilu, P. Habegger, and L. Kühne, Effective bounds for singular units, arXiv:1805.07167 [math.NT], 2018.
Benjamin Braun and Brian Davis, Antichain Simplices, arXiv:1901.01417 [math.CO], 2019.
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
Tibor Burdette and Ian Stewart, Counterexamples to a Conjecture by Alaoglu and Erdős, arXiv:2009.03306 [math.NT], 2020.
Geoffrey Caveney, Jean-Louis Nicolas and Jonathan Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, INTEGERS 11 (2011), #A33.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.
François Clément and Stefan Steinerberger, On the largest singular vector of the Redheffer matrix, arXiv:2502.09489 [math.NT], 2025. See p. 2.
P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants (Text in French), Bulletin de la S. M. F., tome 103 (1975), pp. 65-90.
F. Jokar, On k-layered numbers and some labeling related to k-layered numbers, arXiv:2003.11309 [math.NT], 2020.
Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
A. Morkotun, On the increase of Gronwall function value at the multiplication of its argument by a prime, arXiv preprint arXiv:1307.0083 [math.NT], 2013.
S. Nazardonyavi and S. Yakubovich, Superabundant numbers, their subsequences and the Riemann hypothesis, arXiv preprint arXiv:1211.2147 [math.NT], 2012.
S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434 [math.NT], 2013.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
Walter Nissen, Abundancy : Some Resources.
T. D. Noe, First 500 superabundant numbers.
T. D. Noe, First 1000000 superabundant numbers (21 MB, zipped).
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.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Superabundant Number.
Wikipedia, Superabundant number.

Almost the same as A077006.
The colossally abundant numbers A004490 are a subsequence, as are A023199.
Subsequence of A025487; apart from a(3) = 4 and a(7) = 36, a subsequence of A102750.
Cf. A000203, A002093, A002182.
Cf. A112974 (number of superabundant numbers between colossally abundant numbers).
Cf. A091901 (Robin's inequality), A189686 (superabundant and the reverse of Robin's inequality), A192884 (non-superabundant and the reverse of Robin's inequality).

a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
(* Second program: convert all 8436 terms in b-file into a list of terms: *)
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]}; Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ] (* _Michael De Vlieger, May 08 2018 *)

print1(r=1);forstep(n=2,1e6,2,t=sigma(n,-1);if(t>r,r=t;print1(", "n))) \\ Charles R Greathouse IV, Jul 19 2011

a(n+1) <= 2*a(n). - A.H.M. Smeets, Jul 10 2021

Name edited by Peter Munn, Mar 13 2019

A004490 Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.

Original entry on oeis.org

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, 321253732800, 9316358251200, 288807105787200, 2021649740510400, 6064949221531200, 224403121196654400

Offset: 1

N. J. A. Sloane, Jan 22 2001

nonn

S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 347-407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115.

Amiram Eldar, Table of n, a(n) for n = 1..382 (terms 1..150 from T. D. Noe)
Hirotaka Akatsuka, Maximal order for divisor functions and zeros of the Riemann zeta-function, arXiv:2411.19259 [math.NT], 2024. See p. 4.
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
Kevin Broughan, A Variety of Abundant Numbers, in Equivalents of the Riemann Hypothesis. Cambridge University Press, 2017, pp. 144-164.
Geoffrey 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.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
M. Waldschmidt, From highly composite numbers to transcendental number theory, 2013.
Eric Weisstein's World of Mathematics, Colossally Abundant Number.

A subsequence of A004394 (superabundant numbers).
Cf. A000203, A002201, A073751.
Cf. A002093 (highly abundant numbers), A002182, A005101 (abundant numbers), A006038, A189228 (superabundant numbers that are not colossally abundant).

a(n) = Product_{k=1..n} A073751(k). - Jeppe Stig Nielsen, Nov 28 2021

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.

A112974 Number of superabundant numbers between two consecutive colossally abundant numbers.

Original entry on oeis.org

1, 0, 3, 0, 2, 4, 0, 4, 6, 0, 2, 3, 6, 8, 6, 0, 10, 10, 5, 2, 11, 9, 10, 0, 9, 10, 12, 4, 13, 14, 15, 11, 6, 14, 0, 12, 2, 12, 11, 5, 10, 11, 12, 12, 12, 11, 11, 13, 13, 0, 15, 14, 3, 14, 16, 16, 8, 16, 17, 17, 19, 20, 16, 14, 7, 16, 2, 16, 14, 15, 3, 15, 15, 14, 18, 0, 16, 16, 16, 16, 16, 14

Offset: 1

T. D. Noe, Oct 07 2005

nonn

The colossally abundant numbers are a subset of the superabundant abundant numbers. Is there a formula for a(n) that depends on the two consecutive colossally abundant numbers A004490(n) and A004490(n+1)?

			a(3)=3 because between colossally abundant numbers 12 and 60 there are three superabundant numbers: 24, 36 and 48.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A004490 (colossally abundant numbers), A004394 (superabundant numbers), A189228 (superabundant numbers that are not colossally abundant).

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).

A333655 Highly composite numbers (A002182) that are not superior highly composite numbers (A002201).

Original entry on oeis.org

1, 4, 24, 36, 48, 180, 240, 720, 840, 1260, 1680, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 1081080, 2162160, 2882880, 3603600, 6486480, 7207200, 8648640, 10810800, 14414400

Offset: 1

Iain Fox, Aug 23 2020

nonn

For a number n to be in this sequence, it must have the following conditions be true, where d(n) represents the number of divisors of n (A000005): d(n) > d(k), for all k < n, and there does not exist a number e > 0 such that d(n)/n^e >= d(k)/k^e for k < n and d(n)/n^e > d(k)/k^e for k > n.

This sequence is the same as A189228 until n=12, for which a(12) = 7560 and A189228(12) = 10080.

			4 is in the sequence because it has three factors, more than any preceding number, making it highly composite, but it is not a superior highly composite number.

Iain Fox, Table of n, a(n) for n = 1..9780 (calculated using b-files of A002182 and A002201)
David Terr, Superior Highly Composite Number.
Eric Weisstein's World of Mathematics, Highly Composite Number.
Wikipedia, Highly composite number.
Wikipedia, Superior highly composite number.

Cf. A000005, A189228.
Highly composite numbers: A002182.
Superior highly composite numbers: A002201.

lista(nn)=my(v, w=[1,2,4], r=1, p=primes(primepi(2^log(nn)))); v=setminus(Set(vector(nn, i, prod(n=1, primepi(2^log(i)), p[n]^floor(1/(p[n]^(1/log(i))-1))))), [1]); forstep(x=6, v[#v], 6, if(numdiv(x)>r, r=numdiv(x); w=setunion(w, [x]))); setminus(w, v)

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]]

cp's OEIS Frontend

A004394 Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.

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A004490 Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.

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

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A112974 Number of superabundant numbers between two consecutive colossally abundant numbers.

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

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A333655 Highly composite numbers (A002182) that are not superior highly composite numbers (A002201).

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A340840 Union of the highly composite and superabundant numbers.

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