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A025487
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List giving least integer of each prime signature; also products of primorial numbers A002110.
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237
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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
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OFFSET
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1,2
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COMMENTS
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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".
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001013, A036035, A025488, A051282, A055932, A036041, A061394, A124832.
Equals range of values taken by A046523.
Sequence in context: A048951 A058629 A095810 * A070175 A096850 A062847
Adjacent sequences: A025484 A025485 A025486 * A025488 A025489 A025490
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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David W. Wilson
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EXTENSIONS
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Offset corrected by Matthew Vandermast, Oct 19 2008
Minor correction by Charles R Greathouse IV, Sep 03 2010
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STATUS
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approved
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