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A002093 Highly abundant numbers: numbers n such that sigma(n) > sigma(m) for all m < n.
(Formerly M0553 N0200)
44
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 n such that sum of the even divisors of 2*n 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 have always at least a Goldbach pair of primes. h=p+d. Both observations verified for the first 10000 terms. - David Morales Marciel, Jan 04 2016

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, Highly abundant numbers, Bull. Calcutta Math. Soc., 35, (1943), 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

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 *)

PROG

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

CROSSREFS

Cf. A034091, A000203, A004394, A005153.

The record values are in A034885.

Cf. A193988, A193989 (records for sigma2 and sigma3).

Sequence in context: A139542 A238616 A093717 * A179971 A067069 A100497

Adjacent sequences:  A002090 A002091 A002092 * A002094 A002095 A002096

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

More terms from Jud McCranie, Jul 04 2000

STATUS

approved

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Last modified May 28 16:10 EDT 2016. Contains 273467 sequences.