A034288 -id:A034288 - cp's OEIS frontend

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

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

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

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, ",")))}

A291899 Numbers n such that (pod(n)/tau(n)) > (pod(k)/tau(k)) for all k < n.

Original entry on oeis.org

1, 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, 10080, 12600

Offset: 1

Jaroslav Krizek, Oct 10 2017

nonn

pod(n) = the product of the divisors of n (A007955), tau(n) = the number of the divisors of n (A000005).
Contains all members of A002182 except 2. - Robert Israel, Nov 09 2017
Is this the same as A034288 except for 3? - Georg Fischer, Oct 09 2018
From David A. Corneth, Oct 11 2018: (Start)
Various methods exist to find terms for this sequence, possibly combinable:
- Brute force; checking every positive integer up to some bound.
- Finding terms based on the prime signature.
- Relating to that, the number of divisors.
- Finding terms based on the GCD of some earlier found terms.
- ... (?)
There seems to be a method that helps finding terms < 10^150 for the similar A034287. (End)

			6 is a term because pod(6)/tau(6) = 36/4 = 9 > pod(k)/tau(k) for all k < 6.

David A. Corneth, Table of n, a(n) for n = 1..200 (First 126 terms by Robert Israel)
David A. Corneth, Conjectured first 1171 terms

Cf. A000005, A002182, A034287, A034288, A007955, A120736, A137927, A291186, A067128.

a:=1; S:=[a]; for n in [2..60] do k:=0; flag:= true; while flag do k+:=1; if &*[d: d in Divisors(a)] / #[d: d in Divisors(a)] lt &*[d: d in Divisors(k)] / #[d: d in Divisors(k)] then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;

f:= proc(n) local t; t:= numtheory:-tau(n); simplify(n^(t/2))/t end proc:
N:= 20000: # to get all terms <= N
Res:= NULL: m:= 0:
for n from 1 to N do
  v:= f(n);
  if v > m then Res:= Res, n; m:= v fi
od:
Res; # Robert Israel, Nov 09 2017

With[{s = Array[Times @@ Divisors@ # &, 12600]}, Select[Range@ Length@ s, Function[m, AllTrue[Range[# - 1], m > s[[#]]/DivisorSigma[0, #] &]][s[[#]]/DivisorSigma[0, #]] &]] (* Michael De Vlieger, Oct 10 2017 *)
DeleteDuplicates[Table[{n,Times@@Divisors[n]/DivisorSigma[0,n]},{n,13000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Mar 03 2024 *)

Numbers n such that (A007955(n)/A000005(n)) > (A007955(k)/A000005(k)) for all k < n.

Numbers n such that (A291186(n)/A137927(n)) > (A291186(k)/A137927(k)) for all k < n.

A185166 Number of prime divisors of n (counted with multiplicity) of numbers k such that sum of proper divisors of k exceeds that of all smaller numbers.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 3, 3, 4, 3, 4, 5, 4, 5, 4, 4, 6, 5, 5, 6, 5, 5, 6, 6, 7, 5, 6, 6, 5, 7, 6, 6, 6, 5, 7, 6, 8, 7, 7, 7, 6, 8, 6, 7, 6, 6, 8, 6, 8, 7, 9, 7, 8, 8, 8, 7, 7, 7, 9, 6, 7, 8, 7, 7, 7, 9, 9, 8, 8, 7, 9, 7, 8, 8, 8, 7, 9, 7, 9, 8, 8, 10, 8, 9, 9, 9

Offset: 1

Jonathan Vos Post, Feb 19 2011

nonn

			a(1) = 0 because 1 = A034090(1) has no prime factors.
a(2) = 1 because 2 = A034090(2) has one prime factor, itself.
a(3) = 2 because 4 = A034090(3) = 2^2 has two prime factors (with multiplicity).

Amiram Eldar, Table of n, a(n) for n = 1..1000 (calculated from the b-file at A034090)

Cf. A001222, A034090, A001065, A034091, A034287, A034288.

a(n) = A001222(A034090(n)).

More terms from Amiram Eldar, Aug 30 2019

A290514 Numbers n such that product of odd divisors of n > product of odd divisors of m for all m < n.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 21, 27, 33, 35, 39, 45, 63, 75, 99, 105, 135, 165, 189, 195, 225, 315, 495, 525, 585, 675, 693, 735, 765, 819, 825, 855, 945, 1155, 1365, 1485, 1575, 2205, 2475, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8775, 8925, 9009, 9135, 9405

Offset: 1

Ilya Gutkovskiy, Aug 04 2017

nonn

Numbers n such that A136655(n) > A136655(m) for all m < n.

Cf. A034287, A034288, A136655, A174572, A182469.

mx = 0; t = {}; Do[u = Product[d, {d, Select[Divisors[n], OddQ[#] &]}]; If[u > mx, mx = u; AppendTo[t, n]], {n, 9500}]; t

cp's OEIS Frontend

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

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

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A131984 Where records occur in A106708.

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A291899 Numbers n such that (pod(n)/tau(n)) > (pod(k)/tau(k)) for all k < n.

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A185166 Number of prime divisors of n (counted with multiplicity) of numbers k such that sum of proper divisors of k exceeds that of all smaller numbers.

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A290514 Numbers n such that product of odd divisors of n > product of odd divisors of m for all m < n.

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