A002093 -id:A002093 - cp's OEIS frontend

A128701 Highly abundant numbers that are not products of consecutive primes with nonincreasing exponents, i.e., that are not of the form n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p.

1, 3, 10, 18, 20, 42, 84, 90, 108, 168, 300, 336, 504, 540, 600, 630, 660, 1008, 1200, 1560, 1620, 1980, 2100, 2340, 2400, 3024, 3120, 3240, 3780, 3960, 4200, 4680, 5880, 6120, 6240, 7920, 8400, 8820

Offset: 1

Ant King, Mar 28 2007

nonn

This is the subsequence of those highly abundant numbers (A002093) that have a different canonical structure to the superabundant numbers (A004394), the colossally abundant numbers (A004490), the highly composite numbers (A002182) and the superior highly composite numbers (A002201).

			As 10 is the third highly abundant number that cannot be expressed as a product of consecutive primes with nonincreasing exponents, then a(3)=10.

Amiram Eldar, Table of n, a(n) for n = 1..8404
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, arXiv:math/0008177 [math.NT], 2000-2001.
Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.
Wikipedia, Highly Abundant Numbers.

Cf. A002093, A004394, A000203, A004490, A002182, A002201, A128699, A128700, A128702.

hadata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,10000}]]; data1=Flatten[Position[hadata1,#,1,1]&/@Union[hadata1]];primefactorlist[1]={1};primefactorlist[k_]:=First[Transpose[FactorInteger[k]]];exponentlist[1]={1};exponentlist[k_]:=Last[Transpose[FactorInteger[k]]];g[k_List]:=If[MemberQ[Table[k[[i]]<= k[[i-1]],{i,1,Length[k]}],False],False,True];h[k_]:=If[primefactorlist[k]==(Prime[ # ]&/@Range[Length[primefactorlist[k]]]),True,False];Select[data1,Or[ ! h[ # ],!g[exponentlist[ # ]]]&]
seq = {1}; sm = 0; Do[f = FactorInteger[n]; p = f[[;; , 1]]; e = f[[;; , 2]]; s = Times @@ ((p^(e + 1) - 1)/(p - 1)); If[s > sm, sm = s; m = Length[p]; If[p[[-1]] != Prime[m] || (m > 1 && ! AllTrue[Differences[e], # <= 0 &]), AppendTo[seq, n]]], {n, 2, 10^4}]; seq (* Amiram Eldar, Jun 18 2019 *)

The highly abundant numbers (A002093) are those values of n for which sigma(n)>sigma(m) for all mA000203(n).

A140522 Numbers for which sigma(n) - 2n exceeds sigma(k) - 2k for all k < n.

Original entry on oeis.org

1, 6, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 168, 180, 240, 300, 336, 360, 420, 480, 540, 600, 660, 720, 840, 1008, 1080, 1200, 1260, 1440, 1680, 2100, 2160, 2520, 3240, 3360, 3780, 3960, 4200, 4620, 4680, 5040, 6300, 6720, 7200, 7560, 8400, 9240, 10080

Offset: 1

J. Lowell, Jul 02 2008

nonn

			72 is the smallest number > 60 with an abundance > the abundance of 60. - _Donovan Johnson_, Jan 20 2012

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A002093 (d=0) and A034090 (d=1).

a = {1}; m = -1; For[n = 2, n < 20000, n++, If[DivisorSigma[1, n] - 2*n > m, m = DivisorSigma[1, n] - 2*n; AppendTo[a, n]]]; a (* Stefan Steinerberger, Aug 04 2008 *)
DeleteDuplicates[Table[{n,DivisorSigma[1,n]-2n},{n,11000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Mar 16 2023 *)

More terms from Stefan Steinerberger, Aug 04 2008

A141037 Numbers n where the sum of all of its divisors <= sqrt(n) exceeds the sum of all the divisors of m <= sqrt(m) for all m

Original entry on oeis.org

1, 4, 9, 12, 16, 24, 30, 36, 60, 72, 90, 120, 144, 180, 240, 336, 360, 420, 480, 504, 600, 630, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3600, 3780, 3960, 4200, 4320, 4620, 5040, 6720, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160

Offset: 1

J. Lowell, Jul 28 2008

nonn

			12 qualifies because it sets a record of 1+2+3=6. (1, 2 and 3 are the divisors of 12 <= sqrt(12).)

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A002093 (all divisors), A034090 (all divisors except n itself).

Cf. A066839, A143837 (records of sums).

lst = {}; s = -1; Do[t = Plus @@ Select[Divisors@n, # <= Sqrt@n &]; If[t > s, AppendTo[lst, n]; s = t], {n, 25199}]; lst (* Robert G. Wilson v, Aug 03 2008 *)

More terms from Robert G. Wilson v, Aug 03 2008

A181309 Highly composite numbers that are not highly abundant numbers.

Original entry on oeis.org

1084045767585249647898720000, 63958700287529729226024480000, 6086309919361329033148489516800, 30431549596806645165742447584000, 241271469053348685089061371928480000

Offset: 1

T. D. Noe, Oct 13 2010

nonn

Numbers in A002182 but not in A002093. These terms are A002182(n) for n=255, 278, 301, 312, 362.

From Matthew Vandermast: Alaoglu and Erdos state on page 463 (just before Theorem 18) that "only a finite number of highly abundant numbers can be highly composite." What is the largest number in the intersection of the two sequences?

			n1 = 1084045767585249647898720000 is not highly abundant because the smaller number
n0 = 1082074775280549193993449600 has a larger sum of divisors:
sigma(n1) = 7737797730196290039762124800
sigma(n0) = 7744678597340808238596096000

T. D. Noe, Table of n, a(n) for n=1..10
L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.

A181562 Primes of the form highly abundant number - 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 41, 47, 59, 71, 83, 89, 107, 167, 179, 239, 359, 419, 479, 503, 599, 659, 719, 839, 1259, 1439, 1559, 1619, 1979, 2099, 2339, 2399, 2879, 3023, 3119, 3359, 3779, 4679, 5039, 5879, 6299, 6719, 7559, 7919, 8819, 9239, 10079, 12239, 13859, 21839, 22679, 35279

Offset: 1

Jonathan Vos Post, Jan 29 2011

Note that this sequence and A181561 have an intersection beginning {2, 3, 5, 7, 11, 17, 19, ...}. This sequence UNION A181561 might be called nearly highly abundant primes. That union begins: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 71, 73, 83, 89, 97, 107, 109, 167, 179, 181, 211, 239, 241, 337, 359, 419, 421, 479, 503, 541, 599, 601, 631, 659, 661, 719, 839, 1009, 1201, 1439, 1559, 1619, 1621, 1979, 1801, 2099} and thus has twin nearly highly abundant prime pairs: {(3,5), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73), (107,109), (179,181), (239,241), (419,421), (599,601), (659,661), (1619,1621), ...}.

			The 55th highly abundant number is 2100; subtract one to get 2099, which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..1588

Cf. A000040, A000203, A002093, A181561.

seq = {}; smax = 0; Do[s = DivisorSigma[1, n]; If[s > smax, smax = s; If[PrimeQ[n - 1], AppendTo[seq, n - 1]]], {n, 1, 10^4}]; seq (* Amiram Eldar, Jun 07 2019 *)

{A002093(i) - 1} INTERSECTION A000040.

{(sigma(n) > sigma(m) for all m < n) - 1} INTERSECTION A000040.

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

A286267 Totient highly abundant numbers: numbers n such that A092693(n)+n > A092693(m)+m for all m < n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 23, 29, 37, 41, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 101, 107, 127, 131, 137, 149, 167, 179, 221, 223, 227, 233, 251, 257, 289, 317, 347, 353, 359, 383, 389, 431, 443, 449, 461, 467, 503, 557, 563, 569, 587, 641, 677, 697

Offset: 1

Amiram Eldar, May 05 2017

nonn

Analogous to A002093 (highly abundant numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).

Amiram Eldar, Table of n, a(n) for n = 1..1000
Paul Loomis and Florian Luca, On totient abundant numbers, Electronic Journal of Combinatorial Number Theory, Vol. 8, #A06 (2008).

Cf. A000396, A002093, A082897, A092693, A286265.

Function[s, Flatten[First@ Position[s, #] & /@ Union@ Rest@ FoldList[Max, 0, s]]]@ Table[(Total@ FixedPointList[EulerPhi, n] - 1), {n, 10^3}] (* Michael De Vlieger, May 06 2017 *)

A332320 Numbers k that are highly norm-abundant in Gaussian integers, i.e., A103230(m) < A103230(k) for all m < k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 15, 18, 20, 26, 30, 50, 60, 70, 78, 90, 130, 150, 170, 180, 210, 260, 270, 330, 390, 510, 630, 780, 870, 910, 990, 1020, 1050, 1110, 1170, 1530, 1890, 1950, 2210, 2340, 2550, 2730, 3510, 4290, 4590, 5070, 5460, 5610, 5850, 6630, 8190, 10530

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

Analogous to highly abundant numbers (A002093), with the norm of the sum of divisors function generalized for Gaussian integers (A103230) instead of the sum of divisors function (A000203).

			The first 6 terms of A103230 are 1, 13, 16, 41, 80, 208, 64, 113, 169, 1040. The record values occur at n = 1, 2, 3, 4, 5, 6, 10, the first 7 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..375
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A000203, A002093, A103228, A103229, A103230, A279254, A332321.

s[n_] := Abs[DivisorSigma[1, n, GaussianIntegers -> True]]^2; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1,10^4}]; seq

A070324 a(n) = Max( sigma(k) : k=1,2,3,...,n ).

Original entry on oeis.org

1, 3, 4, 7, 7, 12, 12, 15, 15, 18, 18, 28, 28, 28, 28, 31, 31, 39, 39, 42, 42, 42, 42, 60, 60, 60, 60, 60, 60, 72, 72, 72, 72, 72, 72, 91, 91, 91, 91, 91, 91, 96, 96, 96, 96, 96, 96, 124, 124, 124, 124, 124, 124, 124, 124, 124, 124, 124, 124, 168, 168, 168, 168, 168, 168

Offset: 1

Benoit Cloitre, May 11 2002

Records give A034885. Where records occur gives A002093. - Omar E. Pol, Apr 23 2020

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of (1/n^2) * Sum_{i=1..n} a(i) for n = 2^(1..33).

Cf. A000203, A002093, A034885.

FoldList[Max, DivisorSigma[1, Range[100]]] (* Amiram Eldar, Dec 27 2024 *)

PARI
```
a(n)=vecmax(vector(n,k, sigma(k)))
```

Limit_{n -> infinity} (1/n^2) * Sum_{i=1..n} a(i) = C = 1.2... . [This formula is wrong. See the graph. - Amiram Eldar, Dec 27 2024]

A128689 Numbers k such that A000118(k) is a record.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 10, 14, 18, 26, 30, 42, 54, 66, 78, 90, 114, 126, 150, 186, 198, 210, 270, 330, 378, 390, 450, 510, 546, 570, 630, 798, 810, 882, 930, 990, 1050, 1170, 1350, 1386, 1470, 1530, 1638, 1650, 1710, 1890, 2250, 2310, 2610, 2730, 2970, 3150, 3510, 3570

Offset: 1

Jonathan Vos Post, Mar 21 2007

nonn

This sequence is equivalent to numbers k such that sigma(k) > sigma(m) for all m < k where m and k are not multiples of 4 as no terms are multiples of 4 and A000118(n) = 8*sigma(n) for nonmultiples of 4. - David A. Corneth, May 19 2023

David A. Corneth, Table of n, a(n) for n = 1..392 (terms <= 10^8)

Cf. A000118 (number of ways of writing n as a sum of 4 squares), A002093, A042968 A128690.

f[n_] := SquaresR[4, n];
Module[{record = -1, n}, Reap[For[n = 0, n <= 2000, n++, If[f[n] > record, record = f[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, May 18 2023 *)
DeleteDuplicates[Table[{n,SquaresR[4,n]},{n,0,3600}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jun 20 2024 *)

Missing term 378 added by Giovanni Resta, Jun 18 2016
More terms from Jean-François Alcover, May 18 2023
More terms from David A. Corneth, May 18 2023

cp's OEIS Frontend

A128701 Highly abundant numbers that are not products of consecutive primes with nonincreasing exponents, i.e., that are not of the form n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p.

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A140522 Numbers for which sigma(n) - 2n exceeds sigma(k) - 2k for all k < n.

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A141037 Numbers n where the sum of all of its divisors <= sqrt(n) exceeds the sum of all the divisors of m <= sqrt(m) for all m

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A181309 Highly composite numbers that are not highly abundant numbers.

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A181562 Primes of the form highly abundant number - 1.

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

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A286267 Totient highly abundant numbers: numbers n such that A092693(n)+n > A092693(m)+m for all m < n.

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A332320 Numbers k that are highly norm-abundant in Gaussian integers, i.e., A103230(m) < A103230(k) for all m < k.

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A070324 a(n) = Max( sigma(k) : k=1,2,3,...,n ).