A004394 - OEIS

%I #177 Jun 12 2024 12:06:25

%S 1,2,4,6,12,24,36,48,60,120,180,240,360,720,840,1260,1680,2520,5040,

%T 10080,15120,25200,27720,55440,110880,166320,277200,332640,554400,

%U 665280,720720,1441440,2162160,3603600,4324320,7207200,8648640,10810800,21621600

%N 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.

%C _Matthew Conroy_ points out that these are different from the highly composite numbers - see A002182. Jul 10 1996

%C With respect to the comment above, neither sequence is subsequence of the other. - _Ivan N. Ianakiev_, Feb 11 2020

%C 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

%C 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

%C 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

%C 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

%C See A166735 for superabundant numbers that are not highly composite, and A189228 for superabundant numbers that are not colossally abundant.

%C Pillai called these numbers "highly abundant numbers of the 1st order". - _Amiram Eldar_, Jun 30 2019

%D R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.

%D J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad. Publ., 2002 (see pp. 19-21).

%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

%H D. Kilminster, <a href="/A004394/b004394.txt">Table of n, a(n) for n = 1..2000</a> (extends to n = 8436 in the comments; first 500 terms from T. D. Noe)

%H A. Akbary and Z. Friggstad, <a href="http://www.jstor.org/stable/40391073">Superabundant numbers and the Riemann hypothesis</a>, Amer. Math. Monthly, 116 (2009), 273-275.

%H L. Alaoglu and P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1944-03.pdf">On highly composite and similar numbers,</a> Trans. Amer. Math. Soc., 56 (1944), 448-469. <a href="http://upforthecount.com/math/errata.html">Errata</a>.

%H Christian Axler, <a href="https://arxiv.org/abs/2406.04018">Inequalities involving the primorial counting function</a>, arXiv:2406.04018 [math.NT], 2024. See p. 12.

%H Yu. Bilu, P. Habegger, and L. Kühne, <a href="https://arxiv.org/abs/1805.07167">Effective bounds for singular units</a>, arXiv:1805.07167 [math.NT], 2018.

%H Benjamin Braun and Brian Davis, <a href="https://arxiv.org/abs/1901.01417">Antichain Simplices</a>, arXiv:1901.01417 [math.CO], 2019.

%H Keith Briggs, <a href="https://projecteuclid.org/euclid.em/1175789744">Abundant numbers and the Riemann Hypothesis</a>, Experimental Math., Vol. 16 (2006), p. 251-256.

%H Tibor Burdette and Ian Stewart, <a href="https://arxiv.org/abs/2009.03306">Counterexamples to a Conjecture by Alaoglu and Erdős</a>, arXiv:2009.03306 [math.NT], 2020.

%H Geoffrey Caveney, Jean-Louis Nicolas and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/l33/l33.Abstract.html">Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis</a>, INTEGERS 11 (2011), #A33.

%H G. Caveney, J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1112.6010">On SA, CA, and GA numbers</a>, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012

%H G. Caveney, J.-L. Nicolas and J. Sondow, <a href="http://dx.doi.org/10.1007/s11139-012-9371-0">On SA, CA, and GA numbers</a>, Ramanujan J., 29 (2012), 359-384.

%H P. Erdős and J.-L. Nicolas, <a href="https://doi.org/10.24033/bsmf.1793">Répartition des nombres superabondants (Text in French)</a>, Bulletin de la S. M. F., tome 103 (1975), pp. 65-90.

%H F. Jokar, <a href="https://arxiv.org/abs/2003.11309">On k-layered numbers and some labeling related to k-layered numbers</a>, arXiv:2003.11309 [math.NT], 2020.

%H Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, <a href="https://www.emis.de/journals/JIS/VOL23/Zaikin/zaikin3.html">Enumerating Diagonal Latin Squares of Order Up to 9</a>, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.

%H J. C. Lagarias, <a href="http://arXiv.org/abs/math.NT/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 109 (#6, 2002), 534-543.

%H A. Morkotun, <a href="http://arxiv.org/abs/1307.0083">On the increase of Gronwall function value at the multiplication of its argument by a prime</a>, arXiv preprint arXiv:1307.0083 [math.NT], 2013.

%H S. Nazardonyavi and S. Yakubovich, <a href="http://arxiv.org/abs/1211.2147">Superabundant numbers, their subsequences and the Riemann hypothesis</a>, arXiv preprint arXiv:1211.2147 [math.NT], 2012.

%H S. Nazardonyavi and S. Yakubovich, <a href="http://arxiv.org/abs/1306.3434">Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers</a>, arXiv preprint arXiv:1306.3434 [math.NT], 2013.

%H S. Nazardonyavi and S. Yakubovich, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Nazar/nazar4.html">Extremely Abundant Numbers and the Riemann Hypothesis</a>, Journal of Integer Sequences, 17 (2014), Article 14.2.8.

%H Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources</a>.

%H T. D. Noe, <a href="http://www.sspectra.com/math/SAN.txt">First 500 superabundant numbers</a>.

%H T. D. Noe, <a href="http://www.sspectra.com/math/SAN_1000000.zip">First 1000000 superabundant numbers (21 MB, zipped)</a>.

%H S. Sivasankaranarayana Pillai, <a href="https://web.archive.org/web/20150912090449/http://www.calmathsoc.org/bulletin/article.php?ID=B.1943.35.20">Highly abundant numbers</a>, Bulletin of the Calcutta Mathematical Society, Vol. 35,  No. 1 (1943), pp. 141-156.

%H S. Sivasankaranarayana Pillai, <a href="https://archive.org/details/in.ernet.dli.2015.282686/page/n825">On numbers analogous to highly composite numbers of Ramanujan</a>, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.

%H S. Ramanujan, <a href="http://dx.doi.org/10.1023/A:1009764017495">Highly composite numbers</a>, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.

%H K. P. S. Bhaskara Rao and Yuejian Peng, <a href="https://doi.org/10.1016/j.jnt.2012.09.020">On Zumkeller Numbers</a>, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

%H T. Schwabhäuser, <a href="http://arxiv.org/abs/1308.3678">Preventing Exceptions to Robin's Inequality</a>, arXiv preprint arXiv:1308.3678 [math.NT], 2013.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SuperabundantNumber.html">Superabundant Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Superabundant_number">Superabundant number</a>.

%F a(n+1) <= 2*a(n). - _A.H.M. Smeets_, Jul 10 2021

%t a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]

%t (* Second program: convert all 8436 terms in b-file into a list of terms: *)

%t 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 *)

%o (PARI) 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

%Y Almost the same as A077006.

%Y The colossally abundant numbers A004490 are a subsequence, as are A023199.

%Y Subsequence of A025487; apart from a(3) = 4 and a(7) = 36, a subsequence of A102750.

%Y Cf. A000203, A002093, A002182.

%Y Cf. A112974 (number of superabundant numbers between colossally abundant numbers).

%Y Cf. A091901 (Robin's inequality), A189686 (superabundant and the reverse of Robin's inequality), A192884 (non-superabundant and the reverse of Robin's inequality).

%K nonn,nice

%O 1,2

%A _Matthew Conroy_

%E Name edited by _Peter Munn_, Mar 13 2019