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A002093 Highly abundant numbers: numbers k such that sigma(k) > sigma(m) for all m < k.

%I A002093 M0553 N0200 #86 May 14 2022 11:58:15
%S A002093 1,2,3,4,6,8,10,12,16,18,20,24,30,36,42,48,60,72,84,90,96,108,120,144,
%T A002093 168,180,210,216,240,288,300,336,360,420,480,504,540,600,630,660,720,
%U A002093 840,960,1008,1080,1200,1260,1440,1560,1620,1680,1800,1920,1980,2100
%N A002093 Highly abundant numbers: numbers k such that sigma(k) > sigma(m) for all m < k.
%C A002093 Where record values of sigma(n) occur.
%C A002093 Also record values of A070172: A070172(i) < a(n) for 1 <= i < A085443(n), a(n) = A070172(A085443(n)). - _Reinhard Zumkeller_, Jun 30 2003
%C A002093 Numbers k such that sum of the even divisors of 2*k is a record. - _Arkadiusz Wesolowski_, Jul 12 2012
%C A002093 Conjecture: (a) Every highly abundant number > 10 is practical (A005153). (b) For every integer k there exists A such that k divides a(n) for all n > A. Daniel Fischer proved that every highly abundant number greater than 3, 20, 630 is divisible by 2, 6, 12 respectively. The first conjecture has been verified for the first 10000 terms. - _Jaycob Coleman_, Oct 16 2013
%C A002093 Conjecture: For each term k: (1) Let p be the largest prime less than k (if one exists) and let q be the smallest prime greater than k; then k-p is either 1 or a prime, and q-k is either 1 or a prime. (2) The closest prime number p < k located to a distance d = k-p > 1 is also always at a prime distance. These would mean that the even highly abundant numbers greater than 2 always have at least a Goldbach pair of primes. h=p+d. Both observations verified for the first 10000 terms. - _David Morales Marciel_, Jan 04 2016
%C A002093 Pillai used the term "highly abundant numbers of the r-th order" for numbers with record values of the sum of the reciprocals of the r-th powers of their divisors. Thus highly abundant numbers of the 1st order are actually the superabundant numbers (A004394). - _Amiram Eldar_, Jun 30 2019
%D A002093 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002093 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002093 T. D. Noe, <a href="/A002093/b002093.txt">Table of n, a(n) for n = 1..10000</a>
%H A002093 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A002093 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 A002093 D. Fischer, <a href="http://math.stackexchange.com/a/469199/76533">Proof 2, 6, 12 divides a(n) greater than 3, 20, 630 resp.</a>
%H A002093 S. S. 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 A002093 S. S. 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 A002093 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a> (The RECORDS transform returns both the high-water marks and the places where they occur).
%H A002093 Wikipedia, <a href="http://en.wikipedia.org/wiki/Highly_abundant_number">Highly abundant number</a>
%p A002093 N:= 100: # to get a(1) to a(N)
%p A002093 best:= 0: count:= 0:
%p A002093 for n from 1 while count < N do
%p A002093   s:= numtheory:-sigma(n);
%p A002093   if s > best then
%p A002093     best:= s;
%p A002093     count:= count+1;
%p A002093     A[count]:= n;
%p A002093   fi
%p A002093 od:
%p A002093 seq(A[i],i=1..N);# _Robert Israel_, Jan 20 2016
%t A002093 a={}; k=0; Do[s=DivisorSigma[1,n]; If[s>k, AppendTo[a,n]; k=s], {n,3000}]; a (* _Vladimir Joseph Stephan Orlovsky_, Jul 25 2008 *)
%t A002093 DeleteDuplicates[Table[{n,DivisorSigma[1,n]},{n,100}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* _Harvey P. Dale_, May 14 2022 *)
%o A002093 (PARI) for(n=1,1000,if(sum(i=1,n-1,sign(sigma(n)-sigma(i))) == n-1,print1(n,",")))
%Y A002093 Cf. A034091, A000203, A004394, A005153.
%Y A002093 The record values are in A034885.
%Y A002093 Cf. A193988, A193989 (records for sigma_2 and sigma_3).
%K A002093 nonn,nice
%O A002093 1,2
%A A002093 _N. J. A. Sloane_
%E A002093 Better description from _N. J. A. Sloane_, Apr 15 1997
%E A002093 More terms from _Jud McCranie_, Jul 04 2000