A273018 -id:A273018 - 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

A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor.

Original entry on oeis.org

3, 6, 12, 18, 24, 36, 48, 72, 96, 108

Offset: 1

Vladimir Shevelev, May 13 2016

Theorem. Ramanujan's largely composite numbers (A067128) having the greatest prime divisor p_k = prime(k) do not exceed Product_{2 <= p <= p_k} p^((2*ceiling(log_p(p_(k + 1)) - 1).
Proof. Let N be in A067128 with prime power factorization 2^l_1 * 3^l_2 * ... * p_k^l_k.
First let us show that l_1 <= 2x_1-1 such that 2^x_1 > p_(k+1).
Indeed, consider N_1 = 2^(l_1-x_1)*3^l_2*...*p_k^l_k*p_(k+1).
Since 2^x_1 > p_(k+1) then N_1But d(N_1) > d(N) if l_1 >= 2*x_1, so l_1 <= 2x_1-1.
Analogously we find l_i <= 2x_i-1 if p_i^x_i > p_(k+1), i <= k.
Therefore N <= 2^(2*x_1-1)*3^(2*x_2-1)*...* p_k^(2*x_k-1) and the theorem easily follows.
QED
The inequality of the theorem gives a way to find the full sequence for every p_k. In particular, in case p_k = 2 we have the sequence {2, 4, 8}. For other cases see A273016, A273018.

Cf. A067128, A065119 (the intersection of these two sequences is the present sequence). Cf. also A003586, A273016, A273018.

a = {}; b = {0}; Do[If[# >= Max@ b, AppendTo[a, k] && AppendTo[b, #]] &@ DivisorSigma[0, k], {k, 10^7}]; Select[a, FactorInteger[#][[-1, 1]] == 3 &] (* Michael De Vlieger, May 13 2016 *)

A273057 Largest term in A067128 having greatest prime divisor prime(n).

Original entry on oeis.org

8, 108, 4320, 151200, 3326400, 108108000, 10291881600, 586637251200, 53970627110400, 5869305698256000, 667144414368432000, 26928374543598528000, 3036174229790734032000, 522221967524006253504000, 159538811078583910445472000, 16911113974329894507220032000, 1164048345233041071913645536000, 304315495968066451657424475840000, 28544793521804633165466415833792000, 2026680340048128954748115524199232000

Offset: 1

David A. Corneth and Vladimir Shevelev, May 14 2016

nonn

By the theorem of Vladimir Shevelev mentioned in sequence A273015, such an element exists for each prime.

Cf. A273015, A273016, A273018.

a = {}; b = {0}; Do[If[# >= Max@ b, AppendTo[a, k] && AppendTo[b, #]] &@
DivisorSigma[0, k], {k, 10^7}]; TakeWhile[Table[SelectFirst[Reverse@ a,
FactorInteger[#][[-1, 1]] == Prime@ n &], {n, 6}], # < Max@ a &] (* Michael De Vlieger, May 14 2016 *)

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.

A273235 Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor.

Original entry on oeis.org

3, 10, 17, 28, 27, 43, 44, 69, 68, 58, 97, 97, 125, 164, 201, 185, 162, 254, 263, 313, 491, 434, 466, 417, 309, 358, 510, 633, 935, 1148, 454

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 18 2016

Theorem. The sequence is unbounded.
Proof. Since the sequence of highly composite numbers (A002182) is a subsequence of this sequence, it is sufficient to prove that the number M_n of highly composite numbers with the maximal prime divisor p_n is unbounded. Let N be a large highly composite number. Then for the greatest prime divisor p_N of N we have [Erdos] p_N=O(log N). So for all N<=x, p_N=O(log x).
If M_n=O(1), then the number of all highly composite numbers <=x is O(p_n)=O(log x). However, Erdos [Erdos] proved that this number is more than (log x)^(1+c) for a certain c>0.
So we have a contradiction. This means that M_n and this sequence are unbound. QED

P. Erdős, On Highly composite numbers, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.

Cf. A067128, A273015, A273016, A273018, A273057.

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

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A273015 Ramanujan's largely composite numbers having 3 as the greatest prime divisor.

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A273057 Largest term in A067128 having greatest prime divisor prime(n).

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

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A273235 Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor.

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