User talk:JrandWP/sandbox/Numbers

Latest comment: 4 years ago by Nguyenminh06 in topic Changing of apporved cases?
Documentaion of A025487, which used for this page
Sections Defintion
OFFSET 1,2
COMMENTS All numbers of the form , where , 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 , 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.

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

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.

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

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)

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))

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, A036041, A051466, A061394, A124832, 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 * A279537 A325238 A070175

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

Changing of apporved cases?

edit

I suggest changing that! Nguyenminh06 (talk) 07:05, 11 March 2020 (UTC)Reply