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

Author

Amarnath Murthy, Jan 09 2002

Keywords

• easy
• nonn

Comments

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

Examples

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

Links

• 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.

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Extensions

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