A193988 -id:A193988 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

Cf. A034091, A000203, A004394, A005153.
The record values are in A034885.
Cf. A193988, A193989 (records for sigma_2 and sigma_3).

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

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 *)

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

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

More terms from Jud McCranie, Jul 04 2000

A193989 Numbers n such that sigma_3(n) > sigma_3(k) for all k < n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106

Offset: 1

T. D. Noe, Aug 17 2011

nonn

Where record values of sigma_3(n) occur. RECORDS transform of A001158.

Number of terms up to 10^n for n = 1, 2, ... are 10, 63, 264, 945, 2499, 6365, 12074, 27930, 51660, .... - Charles R Greathouse IV, Sep 02 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A001158, A002093 (highly abundant numbers), A193988.

mx = 0; t = {}; Do[u = DivisorSigma[3, n]; If[u > mx, mx = u; AppendTo[t, n]], {n, 1000}]; t

list(lim)=my(v=List(),r,t); for(n=1,lim, t=sigma(n,3); if(t>r, listput(v,n);r=t)); Vec(v) \\ Charles R Greathouse IV, Sep 02 2015

A055721 Numbers n such that sigma_2(n)/n > sigma_2(k)/k for all k < n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 28, 30, 36, 40, 42, 48, 54, 60, 66, 70, 72, 78, 80, 84, 90, 96, 108, 120, 132, 140, 144, 150, 156, 168, 180, 192, 204, 210, 216, 228, 240, 252, 264, 270, 276, 288, 300, 312, 324, 330, 336, 348, 360, 384, 396, 408

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

sigma_2(n) is the sum of the squares of the divisors of n (A001157).

Amiram Eldar, Table of n, a(n) for n = 1..6000 (terms 1..1000 from Ivan Neretin)

Cf. A002182 (records of sigma_0(n)), A002093 (records of sigma_1(n)), A004394 (records of sigma_1(n)/n), A193988 (records of sigma_2(n)), A208767 (records of sigma_2(n)/n^2).

m:= 0: res:= NULL:
for n from 1 to 500 do
  r:= numtheory:-sigma[2](n)/n;
  if r > m then
    m:= r;
    res:= res, n;
  fi
od:
res; # Robert Israel, Nov 12 2016

a=0; Do[b=DivisorSigma[2, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]

Name edited by Michel Marcus, Nov 12 2016

A381708 a(n) is the smallest nonnegative integer k such that sigma_k(n) > sigma_k(j) for all 1 <= j < n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 1, 2, 1, 3, 0, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 3, 0, 4, 3, 3, 2, 4, 1, 4, 2, 4, 2, 4, 0, 4, 3, 4, 2, 4, 1, 4, 2, 4, 2, 4, 0, 4, 3, 4, 2, 4, 2, 4, 2, 4, 3, 4, 0, 5, 3, 4, 2, 5, 2, 5, 3, 4, 2, 5, 1, 5, 3, 4, 3, 5, 2, 5, 2, 5, 3, 5, 1, 5, 3, 5, 3, 5, 1, 5, 3, 5, 3, 5, 1, 5, 3, 5, 2, 5, 2, 5, 3, 5, 3, 5, 1, 5

Offset: 1

Matthew Conroy, Mar 04 2025

nonn

sigma_k(n) is the sum of the k-th powers of the divisors of n.

a(n) exists since one can prove that for k > n*(log 2 + 1/2 log(n-1)), sigma_k sets a record at n.

			For n = 1, k = 0 is enough so a(1) = 0.
For n = 2, k = 0 works since sigma_0(2) = 2 > 1 = sigma_0(1) so a(2) = 0.
For n = 3, sigma_0(3) = 2 = sigma_0(2), but sigma_1(3) = 1^1+3^1 = 4 > 3 = sigma_1(2) > 1 = sigma_1(1) so a(3) = 1.
For n = 4, sigma_0(4) = 1^0+2^0+4^0 = 3 > 2 = sigma_0(3) = sigma_0(2) > 1 = sigma_0(1) so a(4) = 0.
For n = 5, sigma_0(5) = 2 = sigma_0(2) and sigma_1(5) = 6 < sigma_1(4) = 7 but sigma_2(5) = 26 > sigma_2(4) > sigma_2(3) > sigma_2(2) > sigma_2(1) so a(5) = 2.

Cf. A000203, A001157, A002093, A002182, A098475, A109974, A193988.

check(n,k) =  my(m=0);for(i=1,n-1, my(s=sigma(i,k)); if(s>m,m=s)); if(sigma(n,k)>m,return(1),return(0));
a(n) = my(ii=0); while(!check(n, ii), ii++);  ii;

a(n) = 0 precisely when n is highly composite number A002182.
a(n) = 1 precisely when n is highly abundant A002093 and not highly composite.
a(n) = 2 precisely when n is in A193988 and is not highly composite and is not highly abundant.
a(n) <= m if n < A098475(m). Empirically, it appears that a(A098475(m)) = m+1. - Pontus von Brömssen, Mar 16 2025

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

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A193989 Numbers n such that sigma_3(n) > sigma_3(k) for all k < n.

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A055721 Numbers n such that sigma_2(n)/n > sigma_2(k)/k for all k < n.

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A381708 a(n) is the smallest nonnegative integer k such that sigma_k(n) > sigma_k(j) for all 1 <= j < n.

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