A085988 -id:A085988 - cp's OEIS frontend

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

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

A181821 a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 30, 36, 24, 32, 60, 64, 48, 72, 210, 128, 180, 256, 120, 144, 96, 512, 420, 216, 192, 900, 240, 1024, 360, 2048, 2310, 288, 384, 432, 1260, 4096, 768, 576, 840, 8192, 720, 16384, 480, 1800, 1536, 32768, 4620, 1296, 1080, 1152, 960, 65536

Offset: 1

Matthew Vandermast, Dec 07 2010

nonn

A permutation of A025487. a(n) is the member m of A025487 such that A181819(m) = n. a(n) is also the member of A025487 whose prime signature is conjugate to the prime signature of A108951(n).

If n = Product_i prime(e(i)) with the e(i) weakly decreasing, then a(n) = Product_i prime(i)^e(i). For example, 90 = prime(3) * prime(2) * prime(2) * prime(1), so a(90) = prime(1)^3 * prime(2)^2 * prime(3)^2 * prime(4)^1 = 12600. - Gus Wiseman, Jan 02 2019

			The canonical factorization of 24 is 2^3*3^1. Therefore, p(e(i)) = prime(3)*prime(1)(i.e., A000040(3)*A000040(1)), which equals 5*2 = 10. Since 24 is the smallest integer for which p(e(i)) = 10, a(10) = 24.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181822.
Cf. A046523, A181819, A181820.
Cf. A000040, A001221, A001222, A002110, A056239, A071625, A112798, A118914, A122111, A124859, A182850, A305936, A361808.

a:= n-> (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(sort(map(i->
             numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 05 2018

With[{s = Array[If[# == 1, 1, Times @@ Map[Prime@ Last@ # &, FactorInteger@ #]] &, 2^16]}, Array[First@ FirstPosition[s, #] &, LengthWhile[Differences@ Union@ s, # == 1 &]]] (* Michael De Vlieger, Dec 17 2018 *)
Table[Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,30}] (* Gus Wiseman, Jan 02 2019 *)

A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); }; \\ Antti Karttunen, Dec 10 2018

from math import prod
from sympy import prime, primepi, factorint
def A181821(n): return prod(prime(i)**e for i, e in enumerate(sorted(map(primepi,factorint(n,multiple=True)),reverse=True),1)) # Chai Wah Wu, Sep 15 2023

If A108951(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A108951(n))).
a(A181819(n)) = A046523(n). - [See also A124859]. Antti Karttunen, Dec 10 2018
a(n) = A025487(A361808(n)). - Pontus von Brömssen, Mar 25 2023
a(n) = A108951(A122111(n)). - Antti Karttunen, Sep 15 2023

Definition corrected by Gus Wiseman, Jan 02 2019

A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 60, 8, 2310, 36, 420, 24, 30030, 180, 4620, 120, 510510, 1260, 72, 60060, 16, 900, 840, 9699690, 13860, 360, 1021020, 48, 6300, 9240, 223092870, 180180, 2520, 19399380, 240, 69300, 216, 120120, 6469693230, 1800, 3063060, 144, 44100, 27720, 446185740, 1680, 900900, 1080, 2042040, 200560490130, 12600, 58198140, 32, 720

Offset: 1

Matthew Vandermast, Dec 07 2010

A permutation of the members of A025487.

If integers m and n have conjugate prime signatures, then A001222(m) = A001222(n), A071625(m) = A071625(n), A085082(m) = A085082(n), and A181796(m) = A181796(n).

			A025487(5) = 8 = 2^3 has a prime signature of (3). The partition that is conjugate to (3) is (1,1,1), and the member of A025487 with that prime signature is 30 = 2*3*5 (or 2^1*3^1*5^1).  Therefore, a(5) = 30.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821.

A181825 lists members of A025487 with self-conjugate prime signatures. See also A181823-A181824, A181826-A181827.

f[n_] := Block[{ww, dec}, dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; ww = NestList[Append[#, 1] &, {1}, # - 1] &[-2 + Length@ NestWhileList[NextPrime@ # &, 1, Times @@ {##} <= n &, All] ]; {{{0}}}~Join~Map[Block[{w = #, k = 1}, Sort@ Apply[Join, {{ConstantArray[1, Length@ w]}, If[Length@ # == 0, #, #[[1]]] }] &@ Reap[Do[If[# <= n, Sow[w]; 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[Map[{Times @@ MapIndexed[Prime[First@ #2]^#1 &, #], Times @@ MapIndexed[Prime[First@ #2]^#1 &, Table[LengthWhile[#1, # >= j &], {j, #2}]] & @@ {#, Max[#]}} &, Join @@ f[2310]]][[All, -1]] (* Michael De Vlieger, Oct 16 2018 *)

partitionConj(v)=vector(v[1],i,sum(j=1,#v,v[j]>=i))
primeSignature(n)=vecsort(factor(n)[,2]~,,4)
f(n)=if(n==1, return(1)); my(e=partitionConj(primeSignature(n))~); factorback(concat(Mat(primes(#e)~),e))
A025487=[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];
concat(1, apply(f, A025487)) \\ Charles R Greathouse IV, Jun 02 2016

If A025487(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A025487(n))). a(n) also equals A108951(A181820(n)).

A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.

Original entry on oeis.org

1, 2, 6, 4, 36, 30, 12, 8, 216, 180, 210, 900, 72, 60, 24, 16, 1296, 1080, 1260, 5400, 44100, 2310, 6300, 27000, 432, 360, 420, 1800, 144, 120, 48, 32, 7776, 6480, 7560, 32400, 264600, 13860, 37800, 162000, 9261000, 485100, 30030, 5336100, 1323000, 69300, 189000, 810000, 2592, 2160, 2520, 10800, 88200, 4620, 12600

Offset: 0

Antti Karttunen, Jan 16 2019

A101296(a(n)) gives a permutation of natural numbers.

			The sequence can be represented as a binary tree:
                                      1
                                      |
                   ...................2...................
                  6                                       4
       36......../ \........30                 12......../ \........8
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
   216      180         210    900         72       60         24       16
etc.
Both children are multiples of their common parent, see A323503, A323504 and A323507.
The value of a(n) is computed from the binary expansion of n as follows: Starting from the least significant end of the binary expansion of n (A007088), we record the successive run lengths, subtract one from all lengths except the first one, and use the reversed partial sums of these adjusted values as the exponents of successive primes.
For 11, which is "1011" in base 2, we have run lengths [2, 1, 1] when scanned from the right, and when one is subtracted from all except the first, we have [2, 0, 0], partial sums of which is [2, 2, 2], which stays same when reversed, thus a(11) = 2^2 * 3^2 * 5^2 = 900.
For 13, which is "1101" in base 2, we have run lengths [1, 1, 2] when scanned from the right, and when one is subtracted from all except the first, we have [1, 0, 1], partial sums of which is [1, 1, 2], reversed [2, 1, 1], thus a(13) = 2^2 * 3^1 * 5^1 = 60.
Sequence A227183 is based on the same algorithm.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A000079 (right edge), A000400 (left edge, apart from 2), A005811, A046523, A101296, A227183, A322585, A322825, A323503, A323504, A323507.
Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.
Cf. A005940, A283477, A323505 for other similar trees.

{1}~Join~Array[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ Accumulate@ MapIndexed[Length[#1] - Boole[First@ #2 > 1] &, Split@ Reverse@ IntegerDigits[#, 2]]] &, 54] (* Michael De Vlieger, Feb 05 2020 *)

A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));

a(n) = A046523(a(n)) = A046523(A322825(n)).
A001221(a(n)) = A005811(n).
A001222(a(n)) = A227183(n).
A322585(a(n)) = 1.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Sage

Formula

Extensions

A181821 a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A181822 a(n) = member of A025487 whose prime signature is conjugate to the prime signature of A025487(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A322827 A permutation of A025487: Sequence of least representatives of distinct prime signatures obtained from the run lengths present in the binary expansion of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula