A002093 -id:A002093 - cp's OEIS frontend

A329883 Nonunitary highly abundant numbers: numbers m such that nusigma(m) > nusigma(k) for all k < m, where s(n) is the sum of nonunitary divisors of n (A048146).

1, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 108, 120, 144, 180, 192, 216, 288, 360, 432, 504, 576, 648, 720, 864, 1008, 1080, 1296, 1440, 1728, 1800, 2016, 2160, 2520, 2880, 3024, 3240, 3456, 3528, 3600, 4320, 5040, 5400, 5760, 6048, 6480, 7056, 7200, 8640

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

The corresponding record values are 0, 2, 6, 8, 14, 24, 30, 41, 56, 62, 105, 120, 140, 144, 233, 246, 248, 348, 489, 630, 764, 840, ...

Amiram Eldar, Table of n, a(n) for n = 1..500

The nonunitary version of A002093.

Cf. A048146, A064597, A309141, A329882.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; num = -1; s = {}; Do[nu = nusigma[n]; If[nu > num, num = nu; AppendTo[s, n]], {n, 1, 10^4}]; s

A174572 Highly abundant odd numbers: odd n such that sigma(n) is a record.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 21, 27, 33, 39, 45, 57, 63, 75, 93, 99, 105, 135, 165, 189, 195, 225, 255, 273, 285, 315, 399, 405, 441, 465, 495, 525, 585, 675, 693, 735, 765, 819, 825, 855, 945, 1125, 1155, 1305, 1365, 1485, 1575, 1755, 1785, 1995, 2145, 2205

Offset: 1

T. D. Noe, Mar 24 2010

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A002093 (highly abundant numbers).

mx=0; t={}; Do[ds=DivisorSigma[1,n]; If[ds>mx, mx=ds; AppendTo[t,n]], {n,1,10^4,2}]; t

A181561 Primes of the form highly abundant number + 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 43, 61, 73, 97, 109, 181, 211, 241, 337, 421, 541, 601, 631, 661, 1009, 1201, 1621, 1801, 2161, 2341, 2521, 3121, 3361, 4201, 4621, 5881, 6121, 6301, 7561, 8821, 9241, 12241, 12601, 13441, 15121, 16381, 18481, 19801, 20161, 21841, 23761, 30241, 35281

Offset: 1

Jonathan Vos Post, Jan 29 2011

			The 52nd highly abundant number is 1800, add one to get 1801 which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..1500

Cf. A000040, A000203, A002093.

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.

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

A337691 a(n) is the least positive integer divisible by exactly n primitive nondeficient numbers (A006039).

Original entry on oeis.org

1, 6, 60, 140, 420, 3780, 17160, 28600, 40040, 138600, 120120, 180180, 300300, 360360, 600600, 1351350, 900900, 4144140, 1801800, 3063060, 5405400, 6126120, 8558550, 7657650, 19399380, 20720700, 17117100, 15315300, 29099070, 30630600, 45945900, 70450380, 91891800, 87297210

Offset: 0

David A. Corneth, Antti Karttunen and Peter Munn, Sep 20 2020

nonn

a(10) starts a run of at least 31 terms divisible by 30030 = 13#, product of primes <= 13.

About 20% of known terms are not divisible by 4 (indices 0, 1, 15, 22, 23, 28, 33, 38, 40, ...). This contrasts with many sequences that require terms to have some higher measure of abundancy (cf. A002093, A004394, A004490), where almost all terms are divisible by 4. The possibility of nontrivial odd terms seems worth considering.

			The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = 6, as the smallest number divisible by exactly 1 primitive nondeficient number.
Table of n, a(n) and the relevant divisors starts:
  n    a(n)   divisors in A006039
  0       1   (none);
  1       6   6;
  2      60   6, 20;
  3     140   20, 28, 70;
  4     420   6, 20, 28, 70;
  5    3780   6, 20, 28, 70, 945;
  6   17160   6, 20, 88, 104, 572, 1430;
  7   28600   20, 88, 104, 550, 572, 650, 1430;
  8   40040   20, 28, 70, 88, 104, 572, 1430, 2002; ...
Note that a(6), a(7), a(8) are 3*5720, 5*5720, 7*5720.

David A. Corneth, Table of n, a(n) for n = 0..43
David A. Corneth, Some more upper bounds on a(n) for n at most 110.

A006039, A337690 are used to define this sequence.
See A000203 and A023196 for definitions of deficient and nondeficient.
Sequences with similar definitions: A091193, A335540, A338405.
Cf. A002093, A004394, A004490.

\\ Code for A337690 given under that entry.
A337691list(search_up_to_n) = { my(m=Map(),lista=List([]),t); for(n=1,search_up_to_n,if(!(n%(2^24)),print1("(",n,")")); t=A337690(n); if(!mapisdefined(m,t), mapput(m,t,n))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v337691 = A337691list(2^27);
A337691(n) = v337691[1+n];

a(n) = min({k integer : k >= 1 and A337690(k) = n}).

A348272 Noninfinitary highly abundant numbers: numbers m such that nisigma(m) > nisigma(k) for all k < m, where nisigma(k) is the sum of noninfinitary divisors of n (A348271).

Original entry on oeis.org

1, 4, 9, 12, 16, 28, 36, 48, 80, 100, 112, 144, 180, 240, 300, 324, 336, 396, 400, 432, 468, 528, 576, 684, 720, 900, 1008, 1200, 1296, 1584, 1872, 2160, 2268, 2304, 2448, 2736, 2880, 3312, 3600, 5040, 6300, 6480, 7056, 7920, 9072, 9360, 10800, 11088, 11520, 12240

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

The corresponding record values are 0, 2, 3, 8, 14, 16, 41, 56, 84, 87, 112, ...

			The first 9 values of A348271(k) for k = 1 to 9 are: 0, 0, 0, 2, 0, 0, 0, 0 and 3. The record values, 0, 2 and 3, occur at 1, 4 and 9, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A348271.
The noninfinitary version of A002093.
Similar sequences: A285614, A292983, A327634, A328134, A329883.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq={}; sm = -1; Do[s1 = s[n];If[s1 > sm, sm= s1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A363658 Positive numbers m where A217854(m) is positive and increases to a record.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 12, 18, 20, 24, 30, 40, 42, 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

Simon Jensen, Jun 13 2023

nonn

(-m)^tau(m) > 0 and (-m)^tau(m) > (-k)^tau(k) for all positive k < m, where tau is the number of divisors function.
There are no squares in this sequence.
It appears that if n > 13, then a(n) = A067128(n). See the link.
Only a finite number of terms in A002093 can also be terms in this sequence. See the link.

			5 is a term since (-5)^tau(5) = (-5)^2 = 25 and 25 > (-k)^tau(k) for k = 1,...,4.

Simon Jensen, Table of n, a(n) for n = 1..135
Simon Jensen, On an extended divisor product summatory function

Cf. A363657, A067128, A000005, A002093, A217854, A224914.

isok(m) = my(x=(-m)^numdiv(m)); if (x>0, for (k=1, m-1, if (x <= (-k)^numdiv(k), return(0))); return(1)); \\ Michel Marcus, Aug 31 2023

A065386 Successive record values of the cototient function (A051953).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 22, 24, 30, 32, 36, 44, 46, 48, 54, 60, 66, 70, 72, 78, 88, 90, 92, 94, 96, 110, 120, 132, 138, 140, 162, 176, 180, 184, 198, 210, 220, 250, 264, 270, 294, 324, 330, 342, 352, 360, 382, 396, 402, 426, 440, 486, 500, 514, 522, 528, 550, 588

Offset: 1

Labos Elemer, Nov 05 2001

nonn

			a(8)=22 because for m = 1...29 the cototient values are all smaller than cototient(30)=22, where 30=A065385(8) and 22 is the 8th term in the sequence of such local records.

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

Cototient(A065385(n)).
A006093 gives similar records for the totient function. A002093, A002182, A015702, A005250 are analogous sequences for other functions.
a(n) = A051953(A065385(n)).
Cf. A051953, A065386, A000010.

a=0; s=0; Do[s = n-EulerPhi[n]; If[s>a, a=s; Print[s]], {n, 1, 10000}]
(* Second program: *)
With[{s = Array[# - EulerPhi@ # &, 10^3]}, Union@ FoldList[Max, s]] (* Michael De Vlieger, Nov 03 2017 *)

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("b065386.txt", n++, " ", c); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 17 2009

A070172 Smallest k such that sigma(k) >= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 6, 6, 6, 6, 6, 8, 8, 8, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 36, 36

Offset: 1

Benoit Cloitre, May 06 2002

Also smallest m to partition n into distinct divisors of m; highly abundant numbers are record values: a(i) < A002093(n) for 1<=i < A085443(n), A002093(n) = a(A085443(n)). - Reinhard Zumkeller, Jun 30 2003

1 followed by A002093(k) appearing A034885(k+1)-A034885(k) times, for k >= 2. - Amiram Eldar, Apr 04 2025

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Amiram Eldar, Plot of a(n)*log(log(n))/n for a sample of values of n <= 5.33*10^28.

Cf. A002093, A085443.

nn=80;With[{s=Table[{n,DivisorSigma[1,n]},{n,nn}]},Transpose[ Flatten[ Table[ Select[s,#[[2]]>=i&,1],{i,nn}],1]][[1]]] (* Harvey P. Dale, Dec 28 2013 *)
seq[lim_] := Module[{han = Cases[Import["https://oeis.org/A002093/b002093.txt", "Table"], {, }][[;; , 2]], hmax, sigma, d}, hmax = han[[-1]]; If[lim > hmax, Print["Error: lim is too large"]; {}, han = Select[han, # <= lim &]; sigma = DivisorSigma[1, han]; d = Join[{1}, Differences[sigma]]; Flatten[Table[han[[i]], {i, 1, Length[han]}, {d[[i]]}]]]]; seq[100] (* Amiram Eldar, Apr 04 2025 *)

PARI
```
for(n=1,150,s=1; while(sigma(s)
				
```

It seems that lim_{n -> oo} a(n)*log(log(n))/n = C = 0.6...

A085443 Where records in A070172 occur.

Original entry on oeis.org

1, 2, 4, 5, 8, 13, 16, 19, 29, 32, 40, 43, 61, 73, 92, 97, 125, 169, 196, 225, 235, 253, 281, 361, 404, 481, 547, 577, 601, 745, 820, 869, 993, 1171, 1345, 1513, 1561, 1681, 1861, 1873, 2017, 2419, 2881, 3049, 3225, 3601, 3845, 4369, 4915, 5041, 5083, 5953, 6046

Offset: 1

Reinhard Zumkeller, Jun 30 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002093, A034885, A070172.

A070172(i) < A002093(n) for 1 <= i < a(n).
A002093(n) = A070172(a(n)).
a(n) = A034885(n-1) + 1 for n >= 2. - Amiram Eldar, Apr 04 2025

More terms from Amiram Eldar, Apr 04 2025

A128700 Highly abundant numbers with an odd divisor sum.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 36, 72, 144, 288, 1800, 3600, 7200

Offset: 1

Ant King, Mar 28 2007

Alaoglu and Erdős showed that 7200 is the largest highly abundant number with all the exponents of its prime factors occurring to powers greater than unity. It follows that the sequence of highly abundant numbers with an odd divisor sum is finite and is bounded above by 7200. Accordingly, this is the complete sequence of such integers.

			The fifth highly abundant number with an odd divisor sum is 16. Hence a(5)=16. [Corrected by _N. J. A. Sloane_, Jan 11 2024 at the suggestion of _Harvey P.Dale_.]

L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469.
Wikipedia, Highly Abundant Numbers.

Cf. A002093, A000203.

hadata1=FoldList[Max,1,Table[DivisorSigma[1,n],{n,2,7200}]]; data1=Flatten[Position[hadata1,#,1,1]&/@Union[hadata1]];Select[data1,OddQ[DivisorSigma[1,# ]] &]

The highly abundant numbers are those integers for which sigma(n) > sigma(m) for all m < n (A002093). This sequence contains those elements of A002093 that have an odd divisor sum.

cp's OEIS Frontend

A329883 Nonunitary highly abundant numbers: numbers m such that nusigma(m) > nusigma(k) for all k < m, where s(n) is the sum of nonunitary divisors of n (A048146).

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A174572 Highly abundant odd numbers: odd n such that sigma(n) is a record.

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A181561 Primes of the form highly abundant number + 1.

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A337691 a(n) is the least positive integer divisible by exactly n primitive nondeficient numbers (A006039).

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A348272 Noninfinitary highly abundant numbers: numbers m such that nisigma(m) > nisigma(k) for all k < m, where nisigma(k) is the sum of noninfinitary divisors of n (A348271).

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A363658 Positive numbers m where A217854(m) is positive and increases to a record.

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A065386 Successive record values of the cototient function (A051953).

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A070172 Smallest k such that sigma(k) >= n.

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