A034287 -id:A034287 - cp's OEIS frontend

A067128 Ramanujan's largely composite numbers, defined to be numbers m such that d(m) >= d(k) for k = 1 to m-1.

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

Amarnath Murthy, Jan 09 2002

This sequence is a subsequence of A034287; are they identical? They match for m up to 1500000.
Identical to A034287 for the 105834 terms less than 10^150.
Every subsequence of terms, having the fixed greatest prime divisor prime(k), k=1,2,..., is finite. For a proof see A273015. The list of these subsequences begins {2,4,8}, {3,6,12,18,24,36,48,72,96,108}, ... - Vladimir Shevelev, May 13 2016
By a result of Erdős (1944), a(n+1) <= 2*a(n): see Erdős link. - David A. Corneth, May 20 2016
It appears that if n > 13, then a(n) = A363658(n). - Simon Jensen, Aug 31 2023
Out of the first 10000 terms of this sequence, 1766 are adjacent to a prime. - Dmitry Kamenetsky, Jul 02 2024

			8 is a term as d(8) = 4 and d(k) <= 4 for k = 1,...,7.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On Highly composite numbers, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.
Jean-Louis Nicolas, Répartition des nombres largement composés, Acta Arithmetica 34 (1979), 379-390.
J.-L. Nicolas and G. Robin, Highly Composite Numbers by Srinivasa Ramanujan, The Ramanujan Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.
Vladimir Shevelev, On Erdős constant, arXiv:1605.08884 [math.NT], 2016.

For n with strictly increasing number of divisors, see A002182; A272314, A273011 (infinitary analog), subsequences A273015, A273016, A273018.

Number of divisors of a(n): A273353.

isA067128 := proc(n)
    local nd,k ;
    nd := numtheory[tau](n) ;
    for k from 1 to n-1 do
        if numtheory[tau](k) > nd then
            return false ;
        end if;
    end do:
    true ;
end proc:
A067128 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA067128(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A067128(n),n=1..60) ; # R. J. Mathar, Apr 15 2024

For[n=1; max=0, True, n++, If[(d=DivisorSigma[0, n])>=max, Print[n]; max=d]]
NestList[Function[last,
  NestWhile[# + 1 &, last + 1,
   DivisorSigma[0, #] < DivisorSigma[0, last] &]], 1, 70] (* Steven Lu, Nov 28 2022 *)

is(n) = my(nd=numdiv(n)); for(k=1, n-1, if(numdiv(k) > nd, return(0))); return(1) \\ Felix Fröhlich, May 22 2016

Edited by Dean Hickerson, Jan 15 2002 and by T. D. Noe, Nov 07 2002

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

Original entry on oeis.org

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

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 *)

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.

A174899 Record values of A007955(m), where A007955(m) = product of divisors of m.

Original entry on oeis.org

1, 2, 3, 8, 36, 64, 100, 1728, 5832, 8000, 331776, 810000, 10077696, 254803968, 46656000000, 139314069504, 351298031616, 531441000000, 782757789696, 1586874322944, 42998169600000000, 634562281237118976, 198359290368000000000, 634033809653760000000000

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

a(n) = A007955(A034287(n)).

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A007955 (product of divisors), A034287 (record indices).

{my(m=0); for(n=i=1, 10^3, my(t=vecprod(divisors(n))); if(t>m, print1(t, ", "); m=t))} \\ Andrew Howroyd, Jan 05 2020

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2020

A273014 Least k such that A067128(k) is divisible by n.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 17, 6, 9, 7, 33, 8, 49, 17, 12, 14, 67, 9, 104, 10, 17, 33, 161, 11, 31, 49, 20, 17, 249, 12, 273, 19, 33, 67, 27, 13, 318, 104, 49, 21, 324, 17, 405, 33, 18, 161, 538, 14, 142, 31, 67, 49, 586, 20, 33, 22, 104, 249, 721, 15, 766, 273, 29, 58, 49, 33, 1127, 67, 161, 27, 1252, 16, 1256, 318, 31, 104, 48, 49

Offset: 1

David A. Corneth, May 13 2016

nonn

			A067128(17) = 84. 84 is divisible by the 7. For any m < 17, A067128(m) is not divisible by 7, so a(7) = 17.

David A. Corneth, Table of n, a(n) for n = 1..178

Cf. A034287, A067128.

a = {}; b = {0}; Do[If[# >= Max@ b, AppendTo[a, k] && AppendTo[b, #]] &@ DivisorSigma[0, k], {k, 10^7}]; Table[k = 1; While[! Divisible[a[[k]], n], k++]; k, {n, 22}] (* Michael De Vlieger, May 15 2016, for more terms use b-file data at A067128 *)

a(20)-a(64) from Peter J. C. Moses, May 13 2016

A273353 Number of divisors of A067128(n).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 6, 6, 6, 8, 8, 9, 10, 12, 12, 12, 12, 12, 12, 16, 16, 18, 20, 20, 24, 24, 24, 24, 24, 24, 24, 24, 24, 30, 32, 32, 36, 36, 40, 40, 48, 48, 48, 48, 48, 48, 48, 48, 60, 64, 64, 72, 72, 72, 80, 80, 84, 90, 96, 96, 96, 96, 96, 96, 96, 96, 96, 100, 108, 120, 120, 120, 128, 128, 144, 144, 144, 144, 144, 160

Offset: 1

David A. Corneth, May 20 2016

nonn

Is a(n + 1) / a(n) ~ 1 for large n?
Every term in this sequence also appears in A002183, where every element of this sequence occurs exactly once.
In A067128 it is asked if A034287 = A067128. If that is the case then this sequence is also the number of divisors of A034287.

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

Cf. A000005, A002182, A002183, A034287, A067128.

s = {}; dmax = 0; Do[d = DivisorSigma[0, n]; If[d >= dmax, AppendTo[s, d]; dmax = d], {n, 1, 10^6}]; s (* Amiram Eldar, Jun 07 2019 *)

is_a067128(n) = my(nd=numdiv(n)); for(k=1, n-1, if(numdiv(k) > nd, return(0))); return(1)
for(n=1, 50000, if(is_a067128(n), print1(numdiv(n), ", "))) \\ Felix Fröhlich, May 24 2016

a(n) = A000005(A067128(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

A273038 Least k such that for all m >= k, A067128(m) is divisible by n.

Original entry on oeis.org

1, 4, 11, 33, 35, 11, 50, 67, 94, 35, 113, 33, 197, 50, 35, 249, 247, 94, 276, 35, 50, 113, 427, 67, 354, 197, 453, 50, 765, 35, 855, 544, 113, 247, 50, 94, 1130, 276, 197, 67, 1274, 50, 1457, 113, 94, 427, 1853, 249, 994, 354, 247, 197, 3433, 453, 113, 67, 276, 765, 3757, 35, 4123, 855, 94, 1703, 197, 113, 4465

Offset: 1

David A. Corneth, May 13 2016

nonn

A proof of the existence of a(n) for all n was given by Vladimir Shevelev, May 14 2016, as follows:
(Start)
I give a proof of the existence of k in new David's sequence A273038: "Least k such that for all m >= k, A067128(m) is divisible by n."
Let us change the notation. Suppose N in A067128 has prime power factorization (PPF) N=2^k_1*...*p_n^k_n, k_n>=1,     (1)
where p_i=prime(i).
From my theorem in A273015 it follows that, when N runs through A067128, p_n in (1) is unbounded and, moreover, tends to infinity, when N tends to infinity.
Let us show that, when N runs through A067128, k_1 is also unbounded.
Indeed, suppose k_1 is bounded. Consider a number N_1 with PPF N_1=2^(k_1+x)*...*p_(n-1)^k_(n-1) such that all powers p^i , i=2,...,n-1, are the same as in (1) and satisfy 2^xThen N_1d(N).
We want (k_1+x+1)*...*(k_(n-1)+1)>(k_1+1)*...*(k_(n-1)+1)*(k_n+1), or k_1+x+1>(k_1+1)*(k_n+1)=k_1*k_n+k_n+k_1+1, or x>(k_1+1)*k_n.
So, by (2),  (k_1+1)*k_nSince by hypothesis k_1 is bounded, for large n we can choose the required x, which gives a contradiction. So k_1 is unbounded.
Moreover, we see that k_1 tends to infinity as log_2(p_n), n=n(N), when N tends to infinity, otherwise (3) again leads to contradiction.
Suppose m=2^m_1*3^m_2*...*p_r^m_r.
We can choose k_1 > m_1. In the same way we prove that k_2 tends to infinity and choose k_2 > m_2,..., and so on. k_r tends to infinity and we choose k_r > m_r.  All k_i , i=1,...,r tend to infinity at least as log_p_r(p_n), n=n(N).
So there exists a large M_m such that for all N from A067128 > M_m, m|N.
(End)

Vladimir Shevelev, Posting to Sequence Fans Mailing List, May 14 2016.

Cf. A034287, A067128, A273014, A273015, A273016, A273018.

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A067128 Ramanujan's largely composite numbers, defined to be numbers m such that d(m) >= d(k) for k = 1 to m-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A131984 Where records occur in A106708.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A174899 Record values of A007955(m), where A007955(m) = product of divisors of m.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A273014 Least k such that A067128(k) is divisible by n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A273353 Number of divisors of A067128(n).

Views

Author