A363658 -id:A363658 - 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

A217854 Product of all divisors of n, positive or negative.

Original entry on oeis.org

-1, 4, 9, -64, 25, 1296, 49, 4096, -729, 10000, 121, 2985984, 169, 38416, 50625, -1048576, 289, 34012224, 361, 64000000, 194481, 234256, 529, 110075314176, -15625, 456976, 531441, 481890304, 841, 656100000000, 961, 1073741824, 1185921

Offset: 1

Franklin T. Adams-Watters, Oct 19 2012

sign

a(n) is negative iff n is a square.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000005, A062758, A158387, A363657, A363658.

a[n_] := (-n)^DivisorSigma[0, n]; Array[a, 33] (* Amiram Eldar, Aug 31 2023 *)

a(n) = (-n)^numdiv(n); \\ Michel Marcus, Aug 31 2023

a(n) = (-n)^tau(n) = (-n)^A000005(n).

a(n) = A158387(n) * A062758(n). - Andrew Howroyd, Aug 31 2023

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

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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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A217854 Product of all divisors of n, positive or negative.

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

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