A067128 -id:A067128 - cp's OEIS frontend

A175143 a(1)=1. a(n) = the smallest integer > a(n-1) such that d(a(n))+d(a(n)+1) > d(a(n-1))+d(a(n-1)+1), where d(m) = the number of divisors of m.

1, 2, 3, 5, 8, 11, 15, 20, 24, 35, 59, 80, 84, 119, 224, 239, 335, 359, 360, 480, 539, 719, 720, 840, 1079, 1259, 1260, 1679, 2519, 4199, 5039, 5040, 6720, 7559, 9360, 10079, 10080, 15119, 20159, 25199, 25200, 27719, 32759, 43680, 50399, 55439, 75599

Offset: 1

Leroy Quet, Feb 24 2010

nonn

Those n where A092405(n) sets records.

Nicolas proved that: (1) Except for a finite number of terms, if k is in this sequence either k or k+1 is a largely composite number (A067128). (2) Except for a finite number of terms if k is a highly composite number (A002182) then k-1 is a term of this sequence. Apparently the only exceptions of (1) are 15, 80, 224, 6720, and 9360, and the only exceptions of (2) are 1, 24, 48, 180, 840, and 45360. - Amiram Eldar, Aug 24 2019

Amiram Eldar, Table of n, a(n) for n = 1..145
Jean-Louis Nicolas, Nombres hautement composés, Acta Arithmetica, Vol. 49 (1988), pp. 395-412, alternative link. See p. 398.

Cf. A000005, A002182, A067128, A092405.

A092405 := proc(n) numtheory[tau](n)+numtheory[tau](n+1) ; end proc: read("transforms") ; a092405 :=[seq(A092405(n),n=1..90000)] ; RECORDS(a092405)[2] ; # R. J. Mathar, Mar 05 2010

d1 = 1; dm = 0; s = {}; Do[d2 = DivisorSigma[0, n]; d = d1 + d2; If[d > dm, dm = d; AppendTo[s, n - 1]]; d1 = d2, {n, 2, 80000}]; s (* Amiram Eldar, Aug 24 2019 *)
smi[n_]:=Module[{k=n+1,ds=DivisorSigma[0,n]+DivisorSigma[0,n+1]},While[ DivisorSigma[ 0,k]+DivisorSigma[0,k+1]<=ds,k++];k]; NestList[smi,1,50] (* Harvey P. Dale, Apr 25 2020 *)

Extended by Ray Chandler, Mar 05 2010

Terms beyond 80 from R. J. Mathar, Mar 05 2010

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

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, ",")))}

A273011 Numbers n such that d_i(n) >= d_i(k) for k = 1 to n-1, where d_i(n) is the number of infinitary divisors of n (A037445).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 24, 30, 40, 42, 54, 56, 60, 66, 70, 72, 78, 84, 88, 90, 96, 102, 104, 105, 108, 110, 114, 120, 168, 210, 216, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 420, 440, 456, 462, 480, 504, 510, 520, 540, 546, 552, 570, 594, 600, 616

Offset: 1

Vladimir Shevelev, May 13 2016

nonn

An infinitary (or Fermi-Dirac) analog of the Ramanujan sequence A067128.
Between the smallest number b_k which is product of k distinct terms of A050376 and b_(k+1) all terms are products of k distinct terms of A050376.
Thus every subsequence of terms, having in Fermi-Dirac factorization a fixed number of distinct factors from A050376, is finite.
These subsequences have cardinalities: 1, 4, 10, 21, 47, ...

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

Cf. A037445, A064547, A050376, A067128.

a = {}; b = {0}; Do[If[# >= Max@b, AppendTo[a, k] && AppendTo[b, #]] &@ If[k == 1, 1, Times @@ Flatten@ Map[2^First@ DigitCount[#, 2] &, FactorInteger[k][[All, 2]]]], {k, 10^3}]; a (* Michael De Vlieger, May 13 2016, after Jean-François Alcover at A037445 *)

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.

A036493 Largest number having binary order n (A029837) and of which the number of divisors is maximal in that range of g(k) = n.

Original entry on oeis.org

1, 2, 4, 8, 12, 30, 60, 120, 240, 504, 840, 1680, 3960, 7560, 15120, 32760, 65520, 131040, 262080, 498960, 997920, 1965600, 3603600, 7207200, 14414400, 32432400, 64864800, 122522400, 245044800, 514594080, 1029188160, 2095133040, 4227022800, 8454045600

Offset: 0

Labos Elemer

nonn

This sequence differs from A036451 only at n = 3, 5, 9, 12, and 15, which are the values of n for which there exists more than one k such that g(k) = n and d(k) has the maximum possible value.

a(n) is the largest term k in A067128 such that log_2(k) <= n. - Jon E. Schoenfield, May 13 2018

			For n = 9, k is in {257, 512}, max(d(k)) = 24 (see A036451); this holds for four different numbers (360, 420, 480, and 504); a(9) = 504 since it is the largest.

Cf. A000005, A029837, A005179, A007416, A036451, A036470, A036484.

{1}~Join~Table[Max@ MaximalBy[Range[2^(n - 1) + 1, 2^n], DivisorSigma[0, #] &], {n, 24}] (* Michael De Vlieger, Aug 01 2017 *)

a(22)-a(24) from Michael De Vlieger, Aug 01 2017

a(25)-a(33) from Jon E. Schoenfield, May 12 2018

A097213 Members of A097212, excluding highly composite numbers (A002182).

Original entry on oeis.org

30, 420, 4620, 9240, 12600, 13860, 18480, 138600, 360360, 1801800, 5405400, 6126120, 12252240, 18378360, 24504480, 30630600, 49008960, 91891800, 232792560, 349188840, 465585120, 931170240, 1163962800, 1745944200

Offset: 1

Matthew Vandermast, Aug 07 2004

nonn

A097212 is the RECORDS transform of A076078, the number of sets of distinct positive integers with a least common multiple of n. A002182 is the RECORDS transform of A000005, the number of divisors of n.

A subsequence of A025487 (first integer of each prime signature) and also of A067128 (largely composite numbers).

Amiram Eldar, Table of n, a(n) for n = 1..171

More terms from Robert G. Wilson v and Matthew Vandermast, Aug 23 2004

More terms from David Wasserman, Dec 27 2007

A244353 Largely composite numbers that are not highly composite.

Original entry on oeis.org

3, 8, 10, 18, 20, 30, 72, 84, 90, 96, 108, 168, 336, 420, 480, 504, 540, 600, 630, 660, 672, 1080, 1440, 2160, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 9240, 12600, 13860, 18480, 30240, 32760, 36960, 37800, 40320, 41580, 42840, 43680, 65520, 75600, 98280, 131040, 138600, 151200, 163800, 196560, 262080, 327600, 360360, 393120, 415800, 443520, 471240, 480480, 491400, 655200, 831600, 942480, 982800, 997920

Offset: 1

Daniel Bishop, Jun 26 2014

nonn

Wikipedia, Highly composite number

Equals A067128 excluding A002182.

Wrong term 1 removed by J. Lowell, Feb 25 2019

A329004 a(n) is the largest index in [n] that maximizes tau.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 12, 12, 12, 12, 18, 18, 20, 20, 20, 20, 24, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 72

Offset: 1

Peter Andrew, Jun 10 2020

nonn

Peter Andrew, Table of n, a(n) for n = 1..20000

Cf. A000005, A067128.

a329004 = map fst $ scanl1 (\x y -> maximumBy (comparing snd) [x,y]) $ zip [1..] a000005

a:= proc(n) option remember; uses numtheory; `if`(n=1, 1,
      (t-> `if`(tau(n)Alois P. Heinz, Jun 11 2020

dmax = 0; nmax = 1; seq = {}; Do[If[(d = DivisorSigma[0, n]) >= dmax, dmax = d; nmax = n]; AppendTo[seq, nmax], {n, 1, 72}]; seq (* Amiram Eldar, Jun 11 2020 *)

a(n) = n <=> n in { A067128 }. - Alois P. Heinz, Jun 11 2020

A363657 Numbers m where A217854(m) is a record minimum.

Original entry on oeis.org

1, 4, 9, 16, 36, 100, 144, 324, 400, 576, 900, 1764, 2304, 3600, 7056, 8100, 14400, 28224, 32400, 44100, 57600, 108900, 112896, 129600, 176400, 396900, 435600, 518400, 608400, 705600, 1587600, 2822400, 5336100, 6350400, 14288400, 15681600, 17640000, 21344400

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.
All terms are squares.
It is conjectured that if m is a term, then abs((-m)^tau(m)) <= abs((-k)^tau(k)) for some k < m. See the link.

			9 is a term since (-9)^tau(9) = (-9)^3 = -729 and -729 < (-k)^tau(k) for k = 1,...,8.
25 is not a term since (-25)^tau(5) = (-25)^3 = -15625 > (-16)^tau(16) = (-16)^5 = -1048576 and 16 < 25.

Simon K. Jensen, On an extended divisor product summatory function

Cf. A363658, A000290, A067128, A000005, A217854, A224914.

isok(m) = my(x=(-m)^numdiv(m)); for (k=1, m-1, if (x >= (-k)^numdiv(k), return(0))); return(1); \\ Michel Marcus, Jun 18 2023

cp's OEIS Frontend

A175143 a(1)=1. a(n) = the smallest integer > a(n-1) such that d(a(n))+d(a(n)+1) > d(a(n-1))+d(a(n-1)+1), where d(m) = the number of divisors of m.

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

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PARI

A131984 Where records occur in A106708.

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PARI

A273011 Numbers n such that d_i(n) >= d_i(k) for k = 1 to n-1, where d_i(n) is the number of infinitary divisors of n (A037445).

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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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A036493 Largest number having binary order n (A029837) and of which the number of divisors is maximal in that range of g(k) = n.

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A097213 Members of A097212, excluding highly composite numbers (A002182).

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A244353 Largely composite numbers that are not highly composite.

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A329004 a(n) is the largest index in [n] that maximizes tau.

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