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A002093
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Highly abundant numbers: numbers k such that sigma(k) > sigma(m) for all m < k.
(Formerly M0553 N0200)
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76
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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
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OFFSET
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1,2
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COMMENTS
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Where record values of sigma(n) occur.
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
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REFERENCES
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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).
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LINKS
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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].
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).
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MAPLE
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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:
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MATHEMATICA
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DeleteDuplicates[Table[{n, DivisorSigma[1, n]}, {n, 100}], GreaterEqual[#1[[2]], #2[[2]]]&][[All, 1]] (* Harvey P. Dale, May 14 2022 *)
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PROG
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(PARI) for(n=1, 1000, if(sum(i=1, n-1, sign(sigma(n)-sigma(i))) == n-1, print1(n, ", ")))
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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