This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A002093 Highly abundant numbers: numbers n such that sigma(n) > sigma(m) for all m < n. (Formerly M0553 N0200) 56
 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)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 p1 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. 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 sigma_2 and sigma_3). Sequence in context: A093717 A279029 A316886 * A179971 A067069 A100497 Adjacent sequences:  A002090 A002091 A002092 * A002094 A002095 A002096 KEYWORD nonn,nice AUTHOR EXTENSIONS Better description from N. J. A. Sloane, Apr 15 1997 More terms from Jud McCranie, Jul 04 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 24 19:42 EDT 2019. Contains 321448 sequences. (Running on oeis4.)