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A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110. 446
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

Choie et al. (2007) call these "Hardy-Ramanujan integers". - Jean-François Alcover, Aug 14 2014

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

The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 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.

Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".

YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.

Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.

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. Also published in the book Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261.

Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.

L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.

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 & Dec 24 2019: (Start)

A085089(a(n)) = n.

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

A001221(a(n)) = A061395(a(n)) = A061394(n).

A007814(a(n)) = A051903(a(n)) = A051282(n).

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

a(A306802(n)) = A002182(n).

a(n) = A108951(A181815(n)) = A329900(A181817(n)).

If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).

(End)

Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - Amiram Eldar, Oct 20 2020

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

(PARI) upto(Nmax)=vecsort(concat(vector(logint(Nmax, 2), n, select(t->t<=Nmax, if(n>1, [factorback(primes(#p), Vecrev(p))|p<-partitions(n)], [1, 2]))))) \\ M. F. Hasler, Jul 17 2019

(PARI)

\\ For fast generation of large number of terms, use this program:

A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980

A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); while(lista[i] != u, if(2*lista[i] <= u, listput(lista, 2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista, t))); i++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e.

v025487 = A025487list(101);

A025487(n) = v025487[n];

for(n=1, #v025487, print1(A025487(n), ", ")); \\ Antti Karttunen, Dec 24 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))

N = 2310

nmax = 2^floor(log(N, 2))

sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])

# Giuseppe Coppoletta, Jan 26 2015

CROSSREFS

Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.

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 A055932, of A191743 and of A324583.

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, A260633, A266047, A284456, A300357, A304938, A329894, A330687 also A037019 and A330681 (when sorted), possibly also A289132.

Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.

Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.

Sequence in context: A323508 A324850 A095810 * A333964 A279537 A325238

Adjacent sequences:  A025484 A025485 A025486 * A025488 A025489 A025490

KEYWORD

nonn,easy,nice,core,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 October 21 23:47 EDT 2020. Contains 337948 sequences. (Running on oeis4.)