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
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.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
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 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.
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.
(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 *)
(* Alternative: 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
(SageMath)
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
(Python)
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log, primorial
from oeis_sequences.OEISsequences import bisection
def A025487(n):
@lru_cache(maxsize=None)
def g(x, m, j): return sum(g(x//(prime(m)**i), m-1, i) for i in range(j, integer_log(x, prime(m))[0]+1)) if m-1 else max(0, x.bit_length()-j)
def f(x):
c = n-1+x
for k in count(1):
if primorial(k)>x:
break
c -= g(x, k, 1)
return c
return bisection(f, n, n) # Chai Wah Wu, Mar 21 2026
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import prevprime, nextprime, factorint
def A025487_gen(): # generator of terms if first n terms of sequence is desired
h, hset = [1], {1}
while True:
yield (m:=heappop(h))
ps = factorint(m)
for p in ps:
if p == 2 or ps[prevprime(p)]>ps[p]:
mp = m*p
if mp not in hset:
heappush(h, mp)
hset.add(mp)
mp = m*nextprime(max(ps.keys(), default=1))
if mp not in hset:
heappush(h, mp)
hset.add(mp)
CROSSREFS
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
Equals range of values taken by A046523.
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.
KEYWORD
nonn,easy,nice,core
AUTHOR
EXTENSIONS
Offset corrected by Matthew Vandermast, Oct 19 2008
Minor correction by Charles R Greathouse IV, Sep 03 2010
STATUS
approved
