A002093 -id:A002093 - cp's OEIS frontend

A181311 Highly abundant numbers k whose largest prime factor is less than log(k).

8, 16, 24, 36, 48, 72, 96, 108, 144, 180, 216, 240, 288, 300, 360, 480, 540, 600, 720, 960, 1080, 1200, 1260, 1440, 1620, 1680, 1800, 1920, 2100, 2160, 2400, 2520, 2880, 3024, 3240, 3360, 3600, 3780, 4200, 4320, 5040, 5760, 5880, 6300, 6720, 7200, 7560

Offset: 1

T. D. Noe, Oct 13 2010

nonn

Every highly abundant number (A002093) is either in this sequence or A181312. Although it appears that for every n there is at least one prime p such that n*p is another highly abundant number, this conjecture fails for 46846800, the 227th term.

Amiram Eldar, Table of n, a(n) for n = 1..4292

Cf. A002093, A181312.

seq = {}; sm = 0; Do[s = DivisorSigma[1, n]; If[s > sm, sm = s; If[FactorInteger[n][[-1, 1]] < Log[n], AppendTo[seq, n]]], {n, 1, 8000}]; seq (* Amiram Eldar, Jan 14 2022 *)

A181312 Highly abundant numbers k whose largest prime factor is greater than log(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 18, 20, 30, 42, 60, 84, 90, 120, 168, 210, 336, 420, 504, 630, 660, 840, 1008, 1560, 1980, 2340, 3120, 3960, 4620, 4680, 6120, 6240, 7920, 9240, 10920, 11880, 12240, 13860, 16380, 18480, 19800, 21840, 23760, 27720, 32760, 36960

Offset: 1

T. D. Noe, Oct 13 2010

nonn

Every highly abundant number (A002093) is either in this sequence or A181311. Although it appears that dividing any term by its largest prime factor produces another highly abundant number, this conjecture fails for 2942007868800, the 527th term.

Amiram Eldar, Table of n, a(n) for n = 1..5708

Cf. A002093, A181311.

seq = {}; sm = 0; Do[s = DivisorSigma[1, n]; If[s > sm, sm = s; If[FactorInteger[n][[-1, 1]] > Log[n], AppendTo[seq, n]]], {n, 1, 37000}]; seq (* Amiram Eldar, Jan 14 2022 *)

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

A281782 Numbers n such that sum of prime power divisors of n > sum of prime power divisors of m for all m < n.

Original entry on oeis.org

2, 3, 4, 7, 8, 16, 27, 32, 64, 121, 125, 128, 243, 256, 512, 729, 1024, 2048, 4096, 6561, 8192, 15625, 16384, 32761, 32768, 59049, 65536, 117649, 130321, 131072, 177147, 262144, 524287, 524288, 1048576, 1594323, 1953125, 2097152, 4194304, 8388608

Offset: 1

Ilya Gutkovskiy, Apr 14 2017

nonn

Numbers n such that A023889(n) > A023889(m) for all m < n.

Numbers n such that Sum_{p^k|n, p prime, k>=1} p^k > Sum_{p^k|m, p prime, k>=1} p^k for all m < n.

Amiram Eldar, Table of n, a(n) for n = 1..65
Index entries for sequences related to sums of divisors

Cf. A002093, A002182, A023889, A034090, A174572, A246655, A280013.

mx = 0; t = {}; Do[u = DivisorSum[n, # &, PrimePowerQ[#] &]; If[u > mx, mx = u; AppendTo[t, n]], {n, 8500000}]; t

A316886 Where records occur in A299773.

Original entry on oeis.org

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, 672, 720, 840, 960, 1008, 1080, 1200, 1260, 1440, 1560, 1620, 1680, 1800, 1920, 1980

Offset: 1

Omar E. Pol, Jul 15 2018

nonn

First differs from the highly abundant numbers (A002093) at a(41) = 672, while A002093(41) = 720.

			After a(40) = 660 we have that in the sequence A299773 the terms A299773(661)..A299773(671) are less than A299773(660) = 7187172406818511650939943511021032181119077585. The next term greater than A299773(660) is A299773(672) = 7187180892191062904110726467218877665371246875, so a(41) = 672. Note that both 660 and 672 have the same number of divisors (tau(660) = tau(672) = 24) and the same sum of divisors (sigma(660) = sigma(672) = 2016).

Amiram Eldar, Table of n, a(n) for n = 1..64

Cf. A000005, A000041, A000203, A002093, A299773, A316916.

More terms from Amiram Eldar, Aug 22 2019

A316916 Records in A299773.

Original entry on oeis.org

1, 2, 3, 9, 48, 119, 269, 2740, 5218, 24319, 42150, 792335, 4513067, 54782479, 101527454, 2489565187, 204017663873, 2328040254212, 26770510056270, 60003612992726, 246161422312909, 2017680047306542, 498466688538984687, 7548204089604377821, 705600340631647816172, 26237144094556522735561

Offset: 1

Omar E. Pol, Jul 16 2018

nonn

The indices of these records in A299773 first differ from the highly abundant numbers (A002093) at the 41st term, see A316886.

Amiram Eldar, Table of n, a(n) for n = 1..64

Cf. A000041, A000203, A002093, A299773, A316886.

A326712 Numbers with a record sum of divisors, weighted by divisor multiplicity (A168512).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 324, 336, 360, 420, 480, 504, 540, 576, 600, 660, 672, 720, 840, 900, 960, 1008, 1080, 1200, 1260, 1440, 1560, 1620, 1680, 1800, 1920, 1980, 2016

Offset: 1

Amiram Eldar, Oct 18 2019

nonn

The corresponding record values are 1, 3, 4, 9, 12, 19, 30, 41, 42, 44, 64, 72, 75, 102, 134, 170, 208, 226, 237, 264, 294, ...

			The first values of A168512(n) for n=1..8 are {1, 3, 4, 9, 6, 12, 8, 19}. The record values are 1, 3, 4, 9, 12, 19 for 1, 2, 3, 4, 6, 8. Therefore this sequence begins with 1, 2, 3, 4, 6, 8.

Amiram Eldar, Table of n, a(n) for n = 1..600

Cf. A002093, A168512.

s[n_] := 1 + DivisorSum[n, #*IntegerExponent[n, #] &, # > 1 &]; seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 100000}]; seq (* after Michael De Vlieger at A168512 *)

A348629 Nonexponential highly abundant numbers: numbers m such that nesigma(m) > nesigma(k) for all k < m, where nesigma(k) is the sum of nonexponential divisors of n (A160135).

Original entry on oeis.org

1, 6, 10, 12, 18, 24, 30, 42, 48, 54, 60, 78, 84, 90, 96, 120, 168, 192, 210, 240, 270, 312, 330, 360, 384, 420, 480, 630, 672, 840, 960, 1056, 1080, 1248, 1320, 1440, 1560, 1680, 1890, 1920, 2280, 2310, 2400, 2520, 2640, 2688, 3000, 3120, 3240, 3360, 4200, 4320

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

The corresponding record values are 1, 6, 8, 10, 15, 30, 42, 54, 58, 60, 78, ... (see the link for more values).

			The first 6 values of nesigma(k), for k = 1 to 6 are 1, 1, 1, 1, 1 and 6. The record values, 1 and 6, occur at 1 and 6, the first 2 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..577
Amiram Eldar, Table of n, a(n), A160135(a(n)) for n = 1..577

Cf. A160135, A348604.
The nonexponential version of A002093.
Similar sequences: A285614, A292983, A327634, A328134, A329883, A348272.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1; s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; sm = -1; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A355578 Numbers whose sum of 3-smooth divisors sets a new record.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 108, 144, 192, 216, 288, 324, 384, 432, 576, 648, 768, 864, 972, 1152, 1296, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4608, 5184, 5832, 6912, 7776, 8748, 9216, 10368, 11664, 13824, 15552, 17496

Offset: 1

Amiram Eldar, Jul 08 2022

nonn

Numbers m such that A072079(m) > A072079(k) for all k < m.
All the terms are 3-smooth numbers (A003586).
Equivalently, 3-smooth numbers k such that A000203(k) sets a new record.
Analogous to highly abundant numbers (A002093) with 3-smooth numbers only.

			The numbers of 3-smooth divisors of the first 6 positive integers are 1, 3, 4, 7, 1 and 12. The record values, 1, 3, 4 and 12, occur at 1, 2, 3, 4 and 6, the first 5 terms of this sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..11233 (all < 10^60)

Subsequence of A003586.
A355579 is a subsequence.
Cf. A000203, A002093, A072079, A309015.

s[n_] := Module[{e = IntegerExponent[n, {2, 3}], p}, p = {2, 3}^e; If[Times @@ p == n, (2^(e[[1]] + 1) - 1)*(3^(e[[2]] + 1) - 1)/2, 0]]; sm = 0; seq = {}; Do[sn = s[n]; If[sn > sm, sm = sn; AppendTo[seq, n]], {n, 1, 18000}]; seq

lista(nmax) = {my(list = List(), smax = 0, e2, e3, s); for(n = 1, nmax, e2 = valuation(n, 2); e3 = valuation(n, 3); s = if(2^e2 * 3^e3 == n, (2^(e2 + 1) - 1)*(3^(e3 + 1) - 1)/2, 0); if(s > smax, smax = s;  listput(list, n))); Vec(list)};

from sympy import multiplicity as v
from itertools import count, takewhile
def f(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1)//2
def smooth3(lim):
    pows2 = list(takewhile(lambda x: x record: record = v; records.append(argv)
    return records
print(aupto(10**5)) # Michael S. Branicky, Jul 08 2022

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

cp's OEIS Frontend

A181311 Highly abundant numbers k whose largest prime factor is less than log(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A181312 Highly abundant numbers k whose largest prime factor is greater than log(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Maple

Mathematica

PARI

Extensions

A281782 Numbers n such that sum of prime power divisors of n > sum of prime power divisors of m for all m < n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A316886 Where records occur in A299773.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A316916 Records in A299773.

Views

Author

Keywords

Comments

Links

Crossrefs

A326712 Numbers with a record sum of divisors, weighted by divisor multiplicity (A168512).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A348629 Nonexponential highly abundant numbers: numbers m such that nesigma(m) > nesigma(k) for all k < m, where nesigma(k) is the sum of nonexponential divisors of n (A160135).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A355578 Numbers whose sum of 3-smooth divisors sets a new record.

Views