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

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, 840, 960, 1008, 1080, 1200, 1260, 1440, 1560, 1620, 1680, 1800, 1920, 1980, 2100

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Where record values of sigma(n) occur.

Also record values of A070172: A070172(i) < a(n) for 1 <= i < A085443(n), a(n) = A070172(A085443(n)). - Reinhard Zumkeller, Jun 30 2003

Numbers k such that sum of the even divisors of 2*k is a record. - Arkadiusz Wesolowski, Jul 12 2012

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

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

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

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata.
• D. Fischer, Proof 2, 6, 12 divides a(n) greater than 3, 20, 630 resp.
• S. S. 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. S. Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
• N. J. A. Sloane, Transforms (The RECORDS transform returns both the high-water marks and the places where they occur).
• Wikipedia, Highly abundant number

Crossrefs

Cf. A034091, A000203, A004394, A005153.

The record values are in A034885.

Cf. A193988, A193989 (records for sigma_2 and sigma_3).

Programs

Maple

N:= 100: # to get a(1) to a(N)
best:= 0: count:= 0:
for n from 1 while count < N do
  s:= numtheory:-sigma(n);
  if s > best then
    best:= s;
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i],i=1..N);# Robert Israel, Jan 20 2016

Mathematica

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 *)
DeleteDuplicates[Table[{n,DivisorSigma[1,n]},{n,100}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, May 14 2022 *)

PARI

for(n=1,1000,if(sum(i=1,n-1,sign(sigma(n)-sigma(i))) == n-1,print1(n,",")))

Extensions

Better description from N. J. A. Sloane, Apr 15 1997

More terms from Jud McCranie, Jul 04 2000