A034090 -id:A034090 - cp's OEIS frontend

A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106

Offset: 1

N. J. A. Sloane, R. K. Guy

Also total number of parts in all partitions of n into equal parts that do not contain 1 as a part. - Omar E. Pol, Jan 16 2013
Related concepts: If a(n) < n, n is said to be deficient, if a(n) > n, n is abundant, and if a(n) = n, n is perfect. If there is a cycle of length 2, so that a(n) = b and a(b) = n, b and n are said to be amicable. If there is a longer cycle, the numbers in the cycle are said to be sociable. See examples. - Juhani Heino, Jul 17 2017
Sum of the smallest parts in the partitions of n into two parts such that the smallest part divides the largest. - Wesley Ivan Hurt, Dec 22 2017
a(n) is also the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that do not contain k as a part (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 23 2019
Fixed points are in A000396. - Alois P. Heinz, Mar 10 2024

			x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 7*x^8 + 4*x^9 + 8*x^10 + x^11 + ...
For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
Related concepts: (Start)
From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
Sociable numbers: 12496 -> 14288 -> 15472 -> 14536 -> 14264 -> 12496. See A122726. (End)
For n = 10 the sum of the divisors of 10 that are less than 10 is 1 + 2 + 5 = 8. On the other hand, the partitions of 10 into equal parts that do not contain 1 as a part are [10], [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8. - _Omar E. Pol_, Nov 24 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
H. J. J. te Riele, Perfect numbers and aliquot sequences, pp. 77-94 in J. van de Lune, ed., Studieweek "Getaltheorie en Computers", published by Math. Centrum, Amsterdam, Sept. 1980.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 91.

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].
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Henry Bottomley, Illustration of initial terms
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
Don Coppersmith, An answer to the problem of Don Saari, 1987.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
P. Pollack and C. Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26; errata.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., to appear (2014).
Primefan, Sums of Restricted Divisors for n=1 to 1000
F. Richman, Aliquot series: Abundant, deficient, perfect
Eric Weisstein's World of Mathematics, Restricted Divisor Function
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for "core" sequences

Least inverse: A070015, A359132.
Values taken: A078923, values not taken: A005114.
Records: A034090, A034091.
First differences: A053246, partial sums: A153485.
a(n) = n - A033879(n) = n + A033880(n). - Omar E. Pol, Dec 30 2013
Row sums of A141846 and of A176891. - Gary W. Adamson, May 02 2010
Row sums of A176079. - Mats Granvik, May 20 2012
Alternating row sums of A231347. - Omar E. Pol, Jan 02 2014
a(n) = sum (A027751(n,k): k = 1..A000005(n)-1). - Reinhard Zumkeller, Apr 05 2013
For n > 1: a(n) = A240698(n,A000005(n)-1). - Reinhard Zumkeller, Apr 10 2014
A134675(n) = A007434(n) + a(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
Cf. A032741, A000203, A048050, A000593, A027750.
Cf. A051953, A051731.
Cf. A037020 (primes), A053868, A053869 (odd and even terms).
Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof).
Cf. A007956 (products of proper divisors).
Cf. A005100, A005101, A000396, A259180, A122726 (related concepts).

a001065 n = a000203 n - n  -- Reinhard Zumkeller, Sep 15 2011

[SumOfDivisors(n)-n: n in [1..100]]; // Vincenzo Librandi, May 06 2015

A001065 := proc(n)
    numtheory[sigma](n)-n ;
end proc:
seq( A001065(n),n=1..100) ;

Table[ Plus @@ Select[ Divisors[ n ], #Zak Seidov, Sep 10 2009 *)
Table[DivisorSigma[1, n] - n, {n, 1, 80}] (* Jean-François Alcover, Apr 25 2013 *)
Array[Plus @@ Most@ Divisors@# &, 80] (* Robert G. Wilson v, Dec 24 2017 *)

numlib::sigma(n)-n$ n=1..81 // Zerinvary Lajos, May 13 2008

{a(n) = if( n==0, 0, sigma(n) - n)} /* Michael Somos, Sep 20 2011 */

from sympy import divisor_sigma
def A001065(n): return divisor_sigma(n)-n # Chai Wah Wu, Nov 04 2022

[sigma(n, 1)-n for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - Michael Somos, Jul 05 2006
a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy, Jun 02 2005
a(n) = A155085(-n). - Michael Somos, Sep 20 2011
Equals inverse Mobius transform of A051953 = A051731 * A051953. Example: a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (0, 1, 1, 2, 1, 4) = (0 + 1 + 1 + 0 + 0 + 4), where A051953 = (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, ...) and (1, 1, 1, 0, 0, 1) = row 6 of A051731 where the 1's positions indicate the factors of 6. - Gary W. Adamson, Jul 11 2008
a(n) = A006128(n) - A220477(n) - n. - Omar E. Pol Jan 17 2013
a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - Wesley Ivan Hurt, Oct 25 2013
Dirichlet g.f.: zeta(s-1)*(zeta(s) - 1). - Ilya Gutkovskiy, Aug 07 2016
a(n) = 1 + A048050(n), n > 1. - R. J. Mathar, Mar 13 2018
Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - N. J. A. Sloane, Dec 21 2019
G.f.: Sum_{k >= 2} x^k/(1 - x^k)^2. Cf. A296955. (This follows from the fact that if g(z) = Sum_{n >= 1} a(n)*z^n and f(z) = Sum_{n >= 1} a(n)*z^(N*n)/(1 - z^n) then f(z) = Sum_{k >= N} g(z^k), taking a(n) = n and N = 2.) - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+1))*(n*q^(3*n+2) - (n + 1)*q^(2*n+1) - (n - 1)*q^(n+1) + n)/((1 - q^n)*(1 - q^(n+1))^2). (In equation 1 in Arndt, after combining the two n = 0 summands to get -t/(1 - t), apply the operator t*d/dt to the resulting equation and then set t = q and x = 1.) - Peter Bala, Jan 22 2021
a(n) = Sum_{d|n} d * (1 - [n = d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021
a(n) = Sum_{i=1..n} ((n-1) mod i) - (n mod i). [See also A176079.] - José de Jesús Camacho Medina, Feb 23 2021

A034287 Numbers whose product of divisors is larger than that of any smaller number.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240

Offset: 1

Erich Friedman

It appears that 2 and 3 are the only terms in this sequence that are not in A034288. - T. D. Noe, Mar 10 2007
Is this the same as A067128?
a(n) = numbers m where record values occur in A007955(m); A007955(m) = product of divisors of m. a(n) = possible values of A174901(m) in increasing order, a(n) = the smallest numbers k such that A007955(k) = A174899(n). - Jaroslav Krizek, Apr 01 2010
Equals A067128 for the 105834 terms less than 10^150.

Amiram Eldar, Table of n, a(n) for n = 1..230 (terms below 10^10, terms 1..161 from T. D. Noe)

Cf. A007955, A007956, A034288, A034090, A034091, A067128.

divProd[n_] := Times @@ Divisors[n]; a[1] = 1; a[n_] := a[n] = Catch[For[dp = divProd[an = a[n - 1]]; an++, True, an++, If[divProd[an] > dp, Throw[an]]]]; Table[a[n], {n, 1, 52}] (* Jean-François Alcover, Feb 01 2013 *)
DeleteDuplicates[Table[{n,Times@@Divisors[n]},{n,10000}],GreaterEqual[#1[[2]],#2[[2]]]&] [[;;,1]] (* Harvey P. Dale, Oct 07 2023 *)

A007955(n)=if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2))
r=0;for(n=1,1e5,t=A007955(n);if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013

More terms from David W. Wilson, Dec 19 2001

A034091 Records for sum of proper divisors function A001065.

Original entry on oeis.org

0, 1, 3, 6, 7, 8, 16, 21, 22, 36, 42, 55, 76, 108, 123, 140, 144, 156, 172, 240, 259, 312, 366, 384, 504, 531, 568, 656, 810, 924, 1032, 1056, 1140, 1260, 1356, 1698, 2040, 2088, 2216, 2520, 2644, 3108, 3474, 3480, 4272, 4572, 4844, 5280, 5304, 5412, 6840

Offset: 1

N. J. A. Sloane, J. Lowell

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)

Cf. A001065, A034090.

With[{list1=DeleteDuplicates[Array[DivisorSigma[1, #]-#&, 3000, 0]]}, list2 = {}; Table[If[list1[[n]]>Max[list2], AppendTo[list2, list1[[n]]]], {n, Length[ list1]}]; list2] (* Harvey P. Dale, Dec 30 2011 *)
t = {0}; n = 1; mx = 0; While[Length[t] < 100, n++; s = DivisorSigma[1, n] - n; If[s > mx, mx = s; AppendTo[t, mx]]]; t (* T. D. Noe, Feb 23 2012 *)
DeleteDuplicates[Array[DivisorSigma[1,#]-#&,3000],GreaterEqual] (* Harvey P. Dale, Aug 20 2022 *)

r=0; for(n=1,1e9, t=sigma(n)-n; if(t>r, r=t; print1(t", "))) \\ Charles R Greathouse IV, Feb 09 2017

More terms from Erich Friedman.

A034288 Product of proper divisors is larger than for any smaller number.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240, 10080

Offset: 1

Erich Friedman

Amiram Eldar, Table of n, a(n) for n = 1..228 (terms below 10^10, terms 1..100 from T. D. Noe)

Cf. A034287, A034090, A034091, A067128, A140999.

Indices of records of A007956.

maxTerm = 10^6; record = 0; Reap[For[n = 1, n <= maxTerm, n++, p = Times @@ Most[Divisors[n]]; If[p > record, record = p; Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Aug 01 2013 *)
DeleteDuplicates[Table[{n,Times@@Most[Divisors[n]]},{n,11000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Nov 21 2024 *)

A280013 Numbers k such that sum of squarefree divisors of k > sum of squarefree divisors of m for all m < k.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 14, 21, 22, 26, 30, 42, 66, 78, 102, 114, 130, 138, 170, 174, 186, 210, 318, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1110, 1218, 1230, 1290, 1410, 1554, 1590, 1722, 1770, 1830, 1974, 2010, 2130, 2190, 2310, 2730, 3390, 3570, 3990, 4290, 4830, 5610

Offset: 1

Ilya Gutkovskiy, Apr 14 2017

nonn

Numbers k such that psi(rad(k)) > psi(rad(m)) for all m < k, where psi() is the Dedekind psi function (A001615) and rad() is the squarefree kernel (A007947).
Numbers k such that Sum_{d|k} mu(d)^2*d > Sum_{d|m} mu(d)^2*d for all m < k, where mu() is the Moebius function (A008683).
All terms are squarefree. - Robert Israel, Apr 19 2017

Robert Israel, Table of n, a(n) for n = 1..500
Index entries for sequences related to sums of divisors

Cf. A001615, A002093, A002182, A007947, A008683, A034090, A048250, A174572.

ssd:= n -> convert(select(numtheory:-issqrfree,numtheory:-divisors(n)),`+`):
M:= 0: A:= NULL:
for n from 1 to 10^5 do
    r:= ssd(n);
    if r > M then M:= r; A:= A, n fi
od:
A; # Robert Israel, Apr 19 2017

mx = 0; t = {}; Do[u = DivisorSum[n, # &, SquareFreeQ[#] &]; If[u > mx, mx = u; AppendTo[t, n]], {n, 6000}]; t

from sympy.ntheory.factor_ import core
from sympy import divisors
def s(n): return sum(list(filter(lambda i: core(i) == i, divisors(n))))
def ok(n):
    m=1
    while ms(m): return False
        m+=1
    return True # Indranil Ghosh, Apr 16 2017

A070318 a(n) = Max_{k=1..n} (sigma(k)-k) where sigma(k)-k is the sum of proper divisors of k.

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 6, 7, 7, 8, 8, 16, 16, 16, 16, 16, 16, 21, 21, 22, 22, 22, 22, 36, 36, 36, 36, 36, 36, 42, 42, 42, 42, 42, 42, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 108, 108, 108, 108, 108, 108, 108, 108, 108, 108

Offset: 1

Benoit Cloitre, May 11 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Plot of (1/n^2) * Sum_{i=1..n} a(i) for n = A034090(1..6524) (the positions of records; generated using the b-file at A034090).
Amiram Eldar, Plot of (1/(n^2*log(log(n)))) * Sum_{i=1..n} a(i) for n = A034090(1..6524) (the positions of records; generated using the b-file at A034090).

Cf. A000203, A001065, A034090.

FoldList[Max, Array[DivisorSigma[1, #] - # &, 100]] (* Amiram Eldar, Aug 04 2024 *)

lista(nmax) = {my(smax = -1); for(n = 1, nmax, smax = max(smax, sigma(n) - n); print1(smax, ", "));} \\ Amiram Eldar, Aug 04 2024

Limit_{n -> oo} (1/n^2) * Sum_{i=1..n} a(i) = C = 0.7... . [It seems that this limit in fact diverges to infinity; see the first plot in the links section. - Amiram Eldar, Aug 04 2024]

Conjecture: Limit_{n -> oo} (1/(n^2*log(log(n)))) * Sum_{i=1..n} a(i) = C = 0.7... . (see the second plot in the links section). - Amiram Eldar, Aug 04 2024

A131984 Where records occur in A106708.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 84, 90, 120, 168, 180, 240, 336, 360, 420, 480, 540, 600, 660, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 4200, 4620, 5040, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160, 25200, 27720

Offset: 1

Klaus Brockhaus, Aug 05 2007

Numbers n such that concatenation of proper divisors of n exceeds that of all smaller numbers. Empty concatenation is regarded as 0.

Sequence has many terms in common with A034090 (numbers n such that sum of proper divisors of n exceeds that of all smaller numbers), A034287 (numbers n such that product of divisors of n is larger than for any number less than n), A034288 (product of proper divisors is larger than for any smaller number), A067128 (Ramanujan's largely composite numbers, defined to be n such that d(n) >= d(k) for k = 1 to n-1).

Klaus Brockhaus, Table of n, a(n) for n=1..144

Cf. A106708, A131983 (records), A034090, A034287, A034288, A067128.

DeleteDuplicates[Table[{n,If[CompositeQ[n],FromDigits[Flatten[IntegerDigits/@Rest[ Most[ Divisors[n]]]]],0]},{n,30000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Apr 27 2024 *)

{map(n) = local(d); d=divisors(n); if(#d<3, 0, d[1]=""; eval(concat(vecextract(d, concat("..", #d-1)))))} {m=28000; r=-1; for(n=1, m, if(r<(a=map(n)), r=a; print1(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

A279088 Numbers k for which sigma(k) - 3k exceeds sigma(j) - 3j for all j < k.

Original entry on oeis.org

1, 120, 180, 240, 360, 720, 840, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4680, 5040, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160, 22680, 25200, 27720, 30240, 32760, 36960, 37800, 40320, 42840, 45360, 50400, 55440, 65520, 75600, 83160

Offset: 1

Jon E. Schoenfield, Jan 29 2017

nonn

Positions of record lows in A033885. - Robert Israel, Jan 30 2017

			240 is in the sequence because sigma(240) - 3*240 = 744 - 720 = 24, and no k < 240 has a value of sigma(k) - 3k this large.

Amiram Eldar, Table of n, a(n) for n = 1..350 (terms 1..125 from Robert Israel)

Cf. A002093 (d=0), A034090 (d=1), and A140522 (d=2).

Cf. A033885.

N = 10^6; % to get all terms <= N
V = 1-3*[1:N];
m = V(1);
A(1) = 1;
for n=2:N
  V(n*[1:N/n]) = V(n*[1:N/n]) + n;
  if V(n) > m
    m = V(n);
    A(end+1) = n;
  end
end
A % Robert Israel, Jan 30 2017

m:= numtheory:-sigma(1) - 3:
count:= 1:
A[1]:= 1:
for n from 2 to 10^6 do
  v:= numtheory:-sigma(n)-3*n;
  if v > m then
     count:= count+1;
     A[count]:= n;
     m:= v;
  fi;
od:
seq(A[i],i=1..count); # Robert Israel, Jan 30 2017

With[{s = Array[DivisorSigma[1, #] - 3 # &, 10^5]}, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]] (* Michael De Vlieger, Dec 16 2017 *)

isok(k) = {my(x = sigma(k) - 3*k); for (j=1, k-1, if (sigma(j) - 3*j > x, return (0));); 1;} \\ Michel Marcus, Jan 30 2017

Duplicate a(2)-a(43) removed from b-file by Andrew Howroyd, Feb 27 2018

cp's OEIS Frontend

A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

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A034287 Numbers whose product of divisors is larger than that of any smaller number.

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A034091 Records for sum of proper divisors function A001065.

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A034288 Product of proper divisors is larger than for any smaller number.

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A280013 Numbers k such that sum of squarefree divisors of k > sum of squarefree divisors of m for all m < k.

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A070318 a(n) = Max_{k=1..n} (sigma(k)-k) where sigma(k)-k is the sum of proper divisors of k.

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A131984 Where records occur in A106708.

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