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A025487 List giving least integer of each prime signature; also products of primorial numbers A002110. 241
1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.

A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010

The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".

LINKS

Will Nicholes and Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (Will Nicholes supplied the first 291 terms.)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.

G. H. Hardy and S. Ramanujan, Asymptotic formulae concerning the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132.

FORMULA

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010

Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012

EXAMPLE

The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

MATHEMATICA

PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *)

PROG

(PARI) isA025487(n)=my(k=valuation(n, 2), t); n>>=k; forprime(p=3, default(primelimit), t=valuation(n, p); if(t>k, return(0), k=t); if(k, n/=p^k, return(n==1))) \\ Charles R Greathouse IV, Jun 10 2011

(PARI) factfollow(n)={local(fm, np, n2);

  fm=factor(n); np=matsize(fm)[1];

  if(np==0, return([2]));

  n2=n*nextprime(fm[np, 1]+1);

  if(np==1||fm[np, 2]<fm[np-1, 2], [n*fm[np, 1], n2], [n2])}

al(n) = {local(r, ms); r=vector(n);

  ms=[1];

  for(k=1, n,

    r[k]=ms[1];

    ms=vecsort(concat(vector(#ms-1, j, ms[j+1]), factfollow(ms[1]))));

  r} /* Franklin T. Adams-Watters, Dec 01 2011 */

(Haskell)

import Data.Set (singleton, fromList, deleteFindMin, union)

a025487 n = a025487_list !! (n-1)

a025487_list = 1 : h [b] (singleton b) bs where

   (_ : b : bs) = a002110_list

   h cs s xs'@(x:xs)

     | m <= x    = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'

     | otherwise = x : h (x:cs) (s  `union` fromList (map (* x) (x:cs))) xs

     where (m, s') = deleteFindMin s

-- Reinhard Zumkeller, Apr 06 2013

CROSSREFS

Cf. A001013, A036035, A025488, A051282, A055932, A036041, A061394, A124832.

Equals range of values taken by A046523.

Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).

Sequence in context: A048951 A058629 A095810 * A070175 A096850 A062847

Adjacent sequences:  A025484 A025485 A025486 * A025488 A025489 A025490

KEYWORD

nonn,easy,nice,changed

AUTHOR

David W. Wilson

EXTENSIONS

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

STATUS

approved

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Last modified September 20 18:52 EDT 2014. Contains 247018 sequences.