A002093 -id:A002093 - cp's OEIS frontend

A004490 Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.

2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, 321253732800, 9316358251200, 288807105787200, 2021649740510400, 6064949221531200, 224403121196654400

Offset: 1

N. J. A. Sloane, Jan 22 2001

nonn

S. Ramanujan, Highly composite numbers, Proc. London Math. Soc., 14 (1915), 347-407. Reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, pp. 78-129. See esp. pp. 87, 115.

Amiram Eldar, Table of n, a(n) for n = 1..382 (terms 1..150 from T. D. Noe)
Hirotaka Akatsuka, Maximal order for divisor functions and zeros of the Riemann zeta-function, arXiv:2411.19259 [math.NT], 2024. See p. 4.
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
Kevin Broughan, A Variety of Abundant Numbers, in Equivalents of the Riemann Hypothesis. Cambridge University Press, 2017, pp. 144-164.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
M. Waldschmidt, From highly composite numbers to transcendental number theory, 2013.
Eric Weisstein's World of Mathematics, Colossally Abundant Number.

A subsequence of A004394 (superabundant numbers).
Cf. A000203, A002201, A073751.
Cf. A002093 (highly abundant numbers), A002182, A005101 (abundant numbers), A006038, A189228 (superabundant numbers that are not colossally abundant).

a(n) = Product_{k=1..n} A073751(k). - Jeppe Stig Nielsen, Nov 28 2021

A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

Original entry on oeis.org

1, 3, 3, 7, 3, 12, 3, 15, 7, 15, 3, 28, 3, 15, 15, 31, 3, 39, 3, 42, 15, 15, 3, 60, 7, 15, 15, 56, 3, 72, 3, 63, 15, 15, 15, 91, 3, 15, 15, 90, 3, 96, 3, 63, 55, 15, 3, 124, 7, 63, 15, 63, 3, 120, 15, 120, 15, 15, 3, 168, 3, 15, 59, 127, 15, 144, 3, 63, 15, 142, 3, 195, 3, 15, 63, 63

Offset: 1

Emeric Deutsch, May 15 2006

nonn

If a(n)=sigma(n) (=sum of the divisors of n =A000203(n); i.e. all numbers from 1 to sigma(n) are sums of distinct divisors of n), then n is called a practical number (A005153). The actual sums obtained from the divisors of n are given in row n of the triangle A119348.
The records appear to occur at the highly abundant numbers, A002093, excluding 3 and 10. For n in A174533, a(n) = sigma(n)-2. - T. D. Noe, Mar 29 2010
The indices of records occur at the highly abundant numbers, excluding 3 and 10, if Jaycob Coleman's conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - Amiram Eldar, Jun 13 2020
Zumkeller numbers A083207 give the positions of even terms in this sequence (likewise, the positions of odd terms in A308605). - Antti Karttunen and Ilya Gutkovskiy, Nov 29 2024

			a(5)=3 because the divisors of 5 are 1 and 5 and all the possible sums: are 1,5 and 6; a(6)=12 because we can form all sums 1,2,...,12 by adding up the terms of a nonempty subset of the divisors 1,2,3,6 of 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Jon Maiga, Computer-generated formulas for A119347, Sequence Machine.
B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics 76 (1954), pp. 779-785.

One less than A308605.
Cf. A000203, A002093, A005153, A027750, A030057, A033630, A093890, A100587, A119348, A193279, A225561, A237290, A378447, A378450, A378451.
Cf. A083207 (positions of even terms).

import Data.List (subsequences, nub)
a119347 = length . nub . map sum . tail . subsequences . a027750_row'
-- Reinhard Zumkeller, Jun 27 2015

with(numtheory): with(linalg): a:=proc(n) local dl,t: dl:=convert(divisors(n),list): t:=tau(n): nops({seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)}) end: seq(a(n),n=1..90);

a[n_] := Total /@ Rest[Subsets[Divisors[n]]] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Jan 27 2018 *)

A119347(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); sum(i=1,poldegree(p),(0Antti Karttunen, Nov 28 2024

A119347(n) = { my(c=[0]); fordiv(n, d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A119347(n):
    c = {0}
    for d in divisors(n,generator=True):
        c |=  {a+d for a in c}
    return len(c)-1 # Chai Wah Wu, Jul 05 2023

For n > 1, 3 <= a(n) <= sigma(n). - Charles R Greathouse IV, Feb 11 2019
For p prime, a(p) = 3. For k >= 0, a(2^k) = 2^(k + 1) - 1. - Ctibor O. Zizka, Oct 19 2023
From Antti Karttunen, Nov 29 2024: (Start)
a(n) = A308605(n)-1.
a(n) = 2*(A237290(n)/A000203(n)) - 1. [Found by Sequence Machine. See A237290.]
a(n) <= A100587(n).
(End)

Definition clarified by Antti Karttunen, Nov 29 2024

A034090 Numbers k whose sum of proper divisors (A001065(k)) exceeds that of all smaller numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 660, 720, 840, 960, 1008, 1080, 1200, 1260, 1440, 1560, 1680, 1980, 2100, 2160, 2340, 2400, 2520, 2880, 3120, 3240

Offset: 1

J. Lowell

The highly abundant numbers A002093 are a subsequence since if sigma(k) - k > sigma(m) - m for all m < n then sigma(k) > sigma(m). - Charles R Greathouse IV, Sep 13 2016

			From _William A. Tedeschi_, Aug 19 2010: (Start)
-- 12: 1+2+3+4+6 = 16
13: 1 = 1
14: 1+2+7 = 10
15: 1+3+5 = 9
16: 1+2+4+8 = 15
17: 1 = 1
-- 18: 1+2+3+6+9 = 21
As 12 had the previous (earliest) highest, it is a term; then since 18 has the new highest, it is a term. (End)
Table of initial values of n, a(n), A034091(n) = f(a(n)), where f(k) = sigma(k)-k = A001065(k):
1, 1, 0
2, 2, 1
3, 4, 3
4, 6, 6
5, 8, 7
6, 10, 8
7, 12, 16
8, 18, 21
9, 20, 22
10, 24, 36
11, 30, 42
12, 36, 55
13, 48, 76
14, 60, 108
15, 72, 123
16, 84, 140
17, 90, 144
18, 96, 156
19, 108, 172
20, 120, 240

Don Reble, Table of n, a(n) for n = 1..6524 (first 372 terms from _T. D. Noe_, terms 373 to 1000 from _Donovan Johnson_, terms 1001 to 2750 from Robert G. Wilson v)

This sequence and A034091 together give the record high points in A001065.
Supersequence of A002093.
Cf. A001065, A034091, A034287, A034288.

A = {}; mx = -1; For[ k = 1, k < 10000, k++, t = DivisorSigma[1, k] - k; If[ t > mx, mx = t; AppendTo[A, k]]]; A (* slightly modified by Robert G. Wilson v, Aug 28 2022 *)
DeleteDuplicates[Table[{n,DivisorSigma[1,n]-n},{n,5000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, Jan 15 2023 *)

r=0; for(n=1,1e6, t=sigma(n)-n; if(t>r, r=t; print1(n", "))) \\ Charles R Greathouse IV, Sep 13 2016

More terms from Erich Friedman

A093036 Number of palindromic divisors of a(n) sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 66, 132, 264, 792, 1848, 2772, 5544, 13332, 14652, 24024, 26664, 72072, 79992, 186648, 205128, 264264, 559944, 792792, 1333332, 2666664, 7279272, 7999992, 13333320, 14666652, 26690664, 29333304, 80071992, 134666532, 269333064, 807999192

Offset: 1

Jason Earls, May 08 2004

Beginning with 132, it appears that all entries are congruent mod 11*12; 11 to produce palindromic divisors and 12 for numerous divisors. - Robert G. Wilson v, May 14 2004
The number of palindromic divisors of a(n) are 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 19, 20, 21, 22, 24, 27, 28, 29, 30, 33, 37, 39, 43, 50, 52, 54, 57, 59, 61, 68, 72, 80, 90.
Every term is of the form Product_{i>=1} A226732(i)^e(i) for e(i) >= 0. - David A. Corneth, Jan 10 2021

Jason Earls, "Palindions," Mathematical Bliss, Pleroma Publications, 2009, pages 115-120. ASIN: B002ACVZ6O. [From Jason Earls, Nov 25 2009]

David A. Corneth, Conjectured next terms

Cf. A087990, A002093, A226732.

palindromicQ[n_, b_:10] := If[FromDigits[Reverse[IntegerDigits[n, b]], b] == n, True, False]; a = 0; Do[c = Count[palindromicQ[ # ] & /@ Divisors[n], True]; If[c > a, Print[n]; a = c], {n, 300000000}] (* Robert G. Wilson v, May 14 2004 with a small modification from Alonso del Arte to permit reuse in many other sequences' programs *)

Edited and extended by Robert G. Wilson v, May 14 2004

a(35)-a(36) from Chai Wah Wu, Jan 21 2021

A030057 Least number that is not a sum of distinct divisors of n.

Original entry on oeis.org

2, 4, 2, 8, 2, 13, 2, 16, 2, 4, 2, 29, 2, 4, 2, 32, 2, 40, 2, 43, 2, 4, 2, 61, 2, 4, 2, 57, 2, 73, 2, 64, 2, 4, 2, 92, 2, 4, 2, 91, 2, 97, 2, 8, 2, 4, 2, 125, 2, 4, 2, 8, 2, 121, 2, 121, 2, 4, 2, 169, 2, 4, 2, 128, 2, 145, 2, 8, 2, 4, 2, 196, 2, 4, 2, 8, 2, 169, 2, 187, 2, 4, 2, 225, 2, 4, 2, 181

Offset: 1

David W. Wilson

a(n) = 2 if and only if n is odd. a(2^n) = 2^(n+1). - Emeric Deutsch, Aug 07 2005
a(n) > n if and only if n belongs to A005153, and then a(n) = sigma(n) + 1. - Michel Marcus, Oct 18 2013
The most frequent values are 2 (50%), 4 (16.7%), 8 (5.7%), 13 (3.2%), 16 (2.4%), 29 (1.3%), 32 (1%), 40, 43, 61, ... - M. F. Hasler, Apr 06 2014
The indices of records occur at the highly abundant numbers, excluding 3 and 10, if Jaycob Coleman's conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - Amiram Eldar, Jun 13 2020

			a(10)=4 because 4 is the least positive integer that is not a sum of distinct divisors (namely 1,2,5 and 10) of 10.

David Wasserman and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from David Wasserman)

Cf. A002093, A005153, A093896, A119347.
Distinct elements form A030058.
Cf. A027750.

a030057 n = head $ filter ((== 0) . p (a027750_row n)) [1..] where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) x = if x < k then 0 else p ks (x - k) + p ks x
-- Reinhard Zumkeller, Feb 27 2012

with(combinat): with(numtheory): for n from 1 to 100 do div:=powerset(divisors(n)): b[n]:=sort({seq(sum(div[i][j],j=1..nops(div[i])),i=1..nops(div))}) od: for n from 1 to 100 do B[n]:={seq(k,k=0..1+sigma(n))} minus b[n] od: seq(B[n][1],n=1..100); # Emeric Deutsch, Aug 07 2005

a[n_] :=  First[ Complement[ Range[ DivisorSigma[1, n] + 1], Total /@ Subsets[ Divisors[n]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 02 2012 *)

from sympy import divisors
def A030057(n):
    c = {0}
    for d in divisors(n,generator=True):
        c |=  {a+d for a in c}
    k = 1
    while k in c:
        k += 1
    return k # Chai Wah Wu, Jul 05 2023

Edited by N. J. A. Sloane, May 05 2007

A034885 Record values of sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 12, 15, 18, 28, 31, 39, 42, 60, 72, 91, 96, 124, 168, 195, 224, 234, 252, 280, 360, 403, 480, 546, 576, 600, 744, 819, 868, 992, 1170, 1344, 1512, 1560, 1680, 1860, 1872, 2016, 2418, 2880, 3048, 3224, 3600, 3844, 4368, 4914, 5040, 5082, 5952, 6045

Offset: 1

Erich Friedman

nonn

RECORDS transform of A000203.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..639 from T. D. Noe)
N. J. A. Sloane, Transforms (The RECORDS transform returns both the high-water marks and the places where they occur).

Cf. A000203, A002093, A034091.

DeleteDuplicates[DivisorSigma[1,Range[5000]],GreaterEqual] (* Harvey P. Dale, Dec 20 2023 *)

a(n) = A000203(A002093(n)).

A285614 Unitary highly abundant numbers: numbers n such that usigma(n) > usigma(m) for all m < n, where usigma(n) = sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 14, 18, 21, 22, 26, 30, 42, 60, 66, 78, 90, 102, 114, 130, 138, 150, 170, 174, 186, 210, 294, 318, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1554, 1590, 1722, 1770, 1830, 1974

Offset: 1

Amiram Eldar, Apr 22 2017

nonn

Corresponds to A002093 (Highly abundant numbers), with usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1, A034448) instead of sigma(n) (sum of divisors, A000203).

Contains many terms of A280013 (sum of squarefree divisors instead of unitary divisors), but not all of them - the first terms of A280013 that are not in this sequence are 16530, 26070, 8734110, 8757210,...

			The first 9 values of usigma(n) for n=1..9 are: 1, 3, 4, 5, 6, 12, 8, 9, 10. usigma(10)=18 is higher than these 9 values, thus 10 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A034448, A002093, A034683, A129499, A280013.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; a = {}; k = 0; Do[s = usigma[n]; If[s > k, AppendTo[a, n]; k = s], {n, 1000}]; a

A234521 Sequence of numbers from A234519 such that A234519(n) > A234519(k) for all k < n.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 112, 120, 126, 132, 144, 150, 156, 160, 168, 180, 192, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 276, 280, 288, 300

Offset: 1

Jaroslav Krizek, Jan 13 2014

nonn

A234519 = natural numbers n sorted by decreasing values of number k(n) = sigma(n)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

Conjecture: a(n) = supersequence of A002093 - highly abundant numbers - numbers n such that sigma(n) > sigma(m) for all m < n.

Cf. A234515, A234516, A234517, A234518, A234519, A234520, A234522, A234523, A234524.

A065385 Numbers m at which value of cototient function (A051953) reaches a new record: cototient(m) > cototient(k) for all k < m.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 102, 108, 114, 120, 126, 132, 138, 144, 150, 168, 180, 198, 204, 210, 240, 252, 264, 270, 294, 300, 330, 360, 378, 390, 420, 450, 462, 480, 504, 510, 540, 546, 570, 600, 630, 660, 690, 714

Offset: 1

Labos Elemer, Nov 05 2001

nonn

For totient values prime numbers give similar records.

			a(8) = 30 because for m = 1...29 the cototient values are all smaller than cototient(30) = 22 = A065386(8) and this is the 8th number at which such a record is reached; analogous sequences are A002093, A002182, A015702 or A005250 for functions other than cototient.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A051953, A065386.

With[{s = Array[# - EulerPhi@ # &, 10^3]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, May 16 2018 *)

r=-1; for(n=1,1000,d=n-eulerphi(n); if(r

{ n=0; x=-1; for (m=1, 10^9, c=m - eulerphi(m); if (c > x, x=c; write("b065385.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 17 2009

a=0; s=0; Do[s=n-EulerPhi[n]; If[s>a, a=s; Print[n]], {n, 1, 10000}]

A292983 Bi-unitary highly abundant numbers: numbers n such that bsigma(n) > bsigma(m) for all m < n, where bsigma is the sum of the bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 21, 22, 24, 30, 40, 42, 48, 54, 66, 72, 78, 88, 96, 120, 160, 168, 210, 216, 240, 264, 312, 330, 360, 378, 384, 408, 456, 480, 600, 648, 672, 840, 1056, 1080, 1320, 1512, 1560, 1680, 1848, 1920, 2040, 2184, 2280, 2640

Offset: 1

Amiram Eldar, Sep 27 2017

nonn

Analogous to highly abundant numbers (A002093) with bi-unitary sigma (A188999) instead of sigma (A000203).

Amiram Eldar, Table of n, a(n) for n = 1..569 (terms below 10^10)

Cf. A002093, A188999, A285614.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; a = {}; bmax = 0; Do[b = bsigma[n]; If[b > bmax, AppendTo[a, n]; bmax = b], {n, 3000}]; a (* after Michael De Vlieger at A188999 *)

cp's OEIS Frontend

A004490 Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.

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A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

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A034090 Numbers k whose sum of proper divisors (A001065(k)) exceeds that of all smaller numbers.

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A093036 Number of palindromic divisors of a(n) sets a new record.

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A030057 Least number that is not a sum of distinct divisors of n.

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A034885 Record values of sigma(n).

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A285614 Unitary highly abundant numbers: numbers n such that usigma(n) > usigma(m) for all m < n, where usigma(n) = sum of unitary divisors of n (A034448).

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