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A025487 List giving least integer of each prime signature; also products of primorial numbers A002110. 330
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".

For all such sequences b for which it holds that b(n) = b(A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019

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 formulas 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

From Antti Karttunen, Jan 18 2019: (Start)

A085089(a(n)) = n.

A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]

a(A101296(n)) = A046523(n).

(End)

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, ...

MAPLE

isA025487 := proc(n)

    local pset, omega ;

    pset := sort(convert(numtheory[factorset](n), list)) ;

    omega := nops(pset) ;

    if op(-1, pset) <> ithprime(omega) then

        return false;

    end if;

    for i from 1 to omega-1 do

        if padic[ordp](n, ithprime(i)) < padic[ordp](n, ithprime(i+1)) then

            return false;

        end if;

    end do:

    true ;

end proc:

A025487 := proc(n)

    option remember ;

    local a;

    if n = 1 then

        1 ;

    else

        for a from procname(n-1)+1 do

            if isA025487(a) then

                return a;

            end if;

        end do:

    end if;

end proc:

seq(A025487(n), n=1..100) ; # R. J. Mathar, May 25 2017

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 *)

(* Second program: generate all terms m <= A002110(n): *)

f[n_] := {{1}}~Join~

  Block[{lim = Product[Prime@ i, {i, n}],

   ww = NestList[Append[#, 1] &, {1}, n - 1], dec},

   dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];

   Map[Block[{w = #, k = 1},

      Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],

        Product[Prime@ i, {i, Length@ w}] ] &@ Reap[

         Do[

          If[# < lim,

             Sow[#]; k = 1,

             If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,

             If[k == 1,

               MapAt[# + 1 &, w, k],

               PadLeft[#, Length@ w, First@ #] &@

                 Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],

           {i, Infinity}] ][[-1]]

] &, ww]]; Sort[Join @@ f@ 13] (* Michael De Vlieger, May 19 2018 *)

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 */

(PARI) is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2], , 4) == f[, 2]} \\ David A. Corneth, Feb 14 2019

(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

(Sage) def sharp_primorial(n): return sloane.A002110(prime_pi(n))

def p(n, k): return sharp_primorial(factor(n)[k][0])^factor(n)[k][1];

N=2310; nmax=2^floor(log(N, 2)); sorted([k for k in [prod(p(n, k) for k in range (0, len(factor(n)))) for n in (1..nmax)] if k<=N]) # Giuseppe Coppoletta, Jan 26 2015

CROSSREFS

Cf. A025488, A051282, A055932, A036041, A061394, A124832, A166469, A322584, A322585 (characteristic function).

Cf. A085089, A101296 (left inverses).

Equals range of values taken by A046523.

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

Subsequence of A191743.

Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A266047, A284456, A300357, A304938, also A037019 (when sorted), possibly also A289132.

Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827.

Sequence in context: A323508 A324850 A095810 * A279537 A070175 A096850

Adjacent sequences:  A025484 A025485 A025486 * A025488 A025489 A025490

KEYWORD

nonn,easy,nice,core

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 March 23 14:17 EDT 2019. Contains 321431 sequences. (Running on oeis4.)