A247451 -id:A247451 - cp's OEIS frontend

A307107 a(n) = A025487(n)/A247451(n).

1, 1, 2, 1, 4, 2, 8, 4, 1, 16, 6, 8, 2, 32, 12, 16, 4, 64, 24, 6, 32, 1, 36, 8, 128, 48, 12, 64, 2, 72, 16, 256, 96, 24, 128, 4, 144, 30, 32, 512, 36, 192, 6, 216, 48, 256, 8, 288, 60, 64, 1024, 72, 384, 1, 12, 432, 96, 512, 16, 576, 120, 128, 2048, 144, 768, 2

Offset: 1

Michael De Vlieger, Mar 29 2019

Ratio of A025487(n) and the largest primorial that divides A025487(n). The largest primorial that divides A025487(n) is A002110(A001221(n)).
If A025487(n) is a primorial (i.e., in A002110), a(n) = 1.
a(n) is in A025487 by definition of that sequence as a sorted list of products of primorials.
Conjectures:
1. 1 is the most common value in this sequence even though it only pertains to primorials.
2. All terms in A025487 are in this sequence.

			We can represent the prime divisors p with multiplicity of A025487(n) in a chart where the columns pertain to p and the rows multiplicity. In such a chart, A247451(n) is the longest row (marked by "O" below), and a(n) is the product of primes left over (marked by "X") when we eliminate the primes that produce A247451(n).
A025487(9) =       30 = a(9) * A247451(9)
                      = 1 * 30
  1  O O O
     2 3 5
A025487(27) =     360 = a(27) * A247451(27)
                      = 12 * 30
  3  X
  2  X X
  1  O O O
     2 3 5
A025487(183) = 166320 = a(183) * A247451(183)
                      = 72 * 2310
  4  X
  3  X X
  2  X X
  1  O O O O O
     2 3 5 7 11

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001221, A002110, A025487, A061394, A247451.

f[n_] := {{1}}~Join~Block[{lim = Product[Prime@ i, {i, n}], ww = NestList[Append[#, 1] &, {1}, n - 1], g}, g[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++]] &@ g@ 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]]; With[{s = Union@ Flatten@ f@ 6}, Map[#/Product[Prime@ i, {i, PrimeNu@ #}] &, s]]

a(n) = A025487(n)/A002110(A061394(n)) = A025487(n)/A002110(A001221(A025487(n))).

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

Original entry on oeis.org

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

Offset: 1

David W. Wilson

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

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

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (first 291 terms from Will Nicholes)
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.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
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 formulae for 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.

Subsequence of A055932, A191743, and A324583.
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).
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.

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

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

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 *)

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

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

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

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

\\ 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

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

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

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

A061394 Number of distinct prime factors of n-th least prime signature (A025487); also a(n)-th prime is largest prime factor of n-th least prime signature; also a(n)-th primorial number is largest primorial factor of n-th least product of primorial numbers.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 3, 2, 4, 2, 3, 1, 2, 3, 2, 4, 2, 3, 1, 2, 3, 2, 4, 2, 3, 3, 1, 3, 2, 4, 2, 3, 2, 4, 2, 3, 3, 1, 3, 2, 5, 4, 2, 3, 2, 4, 2, 3, 3, 1, 3, 2, 5, 4, 2, 3, 3, 2, 4, 3, 4, 2, 3, 4, 3, 2, 1, 3, 2, 5, 4, 2, 3, 3, 2, 4, 3, 4, 2, 5, 3, 4, 3, 2, 1, 3, 2, 5, 4, 2, 3, 3

Offset: 1

Henry Bottomley, Apr 30 2001

nonn

A002110(a(n)) = A247451(n). - Reinhard Zumkeller, Sep 17 2014

Number of parts of the associated prime signature. - Álvar Ibeas, Nov 01 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002110, A247451, A006530, A061395, A025487, A000040, A051903. A001221 by prime signature.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a061394 = fromJust . (`elemIndex` a002110_list) . a247451
-- Reinhard Zumkeller, Sep 17 2014

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)))
[omega(n) | n <- [1..1000], isA025487(n)]
\\ Or, for older versions:
apply(omega, select(isA025487, [1..1000])) \\ Charles R Greathouse IV, Nov 07 2014

a(n) = A061395(A025487(n)) = A001221(A025487(n)) = A051903(A181822(n)).

A000040(a(n)) = A006530(A025487(n)).

Offset updated by Matthew Vandermast, Nov 08 2008

cp's OEIS Frontend

A307107 a(n) = A025487(n)/A247451(n).

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

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A061394 Number of distinct prime factors of n-th least prime signature (A025487); also a(n)-th prime is largest prime factor of n-th least prime signature; also a(n)-th primorial number is largest primorial factor of n-th least product of primorial numbers.

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