A181826 -id:A181826 - 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

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)).

A212167 Numbers k such that the maximum exponent in its prime factorization is not greater than the number of positive exponents (A051903(k) <= A001221(k)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Matthew Vandermast, May 22 2012

nonn

Union of A212166 and A212168. Includes numerous subsequences that are subsequences of neither A212166 nor A212168.

			40 = 2^3*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although the 1 is often left implicit).  2 is less than the maximal exponent in 40's prime factorization, which is 3. Therefore, 40 does not belong to the sequence. But 10 = 2^1*5^1 and 20 = 2^2*5^1 belong, since the maximal exponents in their prime factorizations are 1 and 2 respectively.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212164. See also A212165.
Subsequences (none of which are subsequences of A212166 or A212168) include A002110, A051451, A129912, A179983, A181826, A181827, A182862, A182863. Includes all members of A003418.
Cf. A001221, A050326, A051903, A188654, A225230.

import Data.List (findIndices)
a212167 n = a212167_list !! (n-1)
a212167_list = map (+ 1) $ findIndices (>= 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

isA212167 := proc(n)
    simplify(A051903(n) <= A001221(n)) ;
end proc:
for n from 1 to 1000 do
    if isA212167(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 06 2021

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] <= Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) <= #e; } \\ Amiram Eldar, Sep 09 2024

A225230(a(n)) >= 0; A050326(a(n)) > 0. - Reinhard Zumkeller, May 03 2013

A181824 Members of A025487 such that A025487(n) <= A181822(n).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 360, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1080, 1152, 1296, 1440, 1536, 1680, 1728, 1920, 2048, 2160, 2304, 2592, 2880, 3072, 3360, 3456, 3600, 3840, 4096

Offset: 1

Matthew Vandermast, Dec 08 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Union of A181823 and A181825.

Cf. A025487, A181822, A181826, A181827.

A181825 Members of A025487 whose prime signature is self-conjugate (as a partition).

Original entry on oeis.org

1, 2, 12, 36, 120, 360, 1680, 5040, 5400, 27000, 36960, 75600, 110880, 378000, 960960, 1587600, 1663200, 2882880, 7938000, 8316000, 32672640, 34927200, 43243200, 98017920, 174636000, 216216000, 277830000, 908107200, 1152597600, 1241560320, 1470268800, 1944810000

Offset: 1

Matthew Vandermast, Dec 08 2010

A025487(n) is included iff A025487(n) = A181822(n).

Closed under the binary operations of GCD and LCM, since a self-conjugate partition of Omega(a(n)) (which the prime signature of these numbers is) is the concatenation of self-conjugate hooks of decreasing size while moving downward and to the right in the Ferrers diagram, and the GCD (or LCM) of two terms a(i) and a(j) is obtained by taking the smaller (or larger, respectively) of the corresponding hooks. For example, GCD(a(8),a(11)) = GCD(5040,36960) = 1680 = a(7), and LCM(a(8),a(11)) = 110880 = a(13). The two binary operations make the set {a(n)} into a lattice order. - Richard Peterson, May 29 2020

			A025487(11) = 36 = 2^2*3^2 has a prime signature of (2,2), which is a self-conjugate partition; hence, 36 is included in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11879 (first 578 terms from Amiram Eldar, terms <= 10^70)
David A. Corneth, PARI-program
Eric Weisstein's World of Mathematics, Self-Conjugate Partition

Cf. A025487, A181822, A303557.
Includes subsequences A006939 and A181555.
See also A181823, A181824, A181826, A181827.

\\ See Corneth link \\ David A. Corneth, Jun 03 2020

a(18)-a(32) from Amiram Eldar, Jan 19 2019

A181823 Members of A025487 such that A025487(n) < A181822(n).

Original entry on oeis.org

4, 8, 16, 24, 32, 48, 64, 72, 96, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1080, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2160, 2304, 2592, 2880, 3072, 3360, 3456, 3600, 3840, 4096, 4320, 4608, 5184, 5760, 6144, 6480

Offset: 1

Matthew Vandermast, Dec 08 2010

nonn

			A025487(5) = 8 and A181822(5) = 30 have the prime signatures (3) and (1,1,1) respectively. 8 is the smaller member of the pair and is therefore included in this sequence.

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

Cf. A025487, A181822, A181824, A181825, A181826, A181827.

A181827 Members of A025487 such that A025487(n) > A181822(n).

Original entry on oeis.org

6, 30, 60, 180, 210, 420, 840, 900, 1260, 1800, 2310, 2520, 4620, 6300, 7560, 9240, 12600, 13860, 18480, 25200, 27720, 30030, 37800, 44100, 55440, 60060, 69300, 83160, 88200, 120120, 138600, 166320, 176400, 180180, 189000, 240240, 264600, 277200

Offset: 1

Matthew Vandermast, Dec 08 2010

nonn

			A025487(9) = 30 and A181822(9) = 8 have the prime signatures (1,1,1) and (3) respectively. 30 is the larger member of the pair and is therefore included in this sequence.

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

Cf. A025487, A181822, A181823, A181824, A181825, A181826.

A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 1260, 2520, 27720, 138600, 360360, 831600, 10810800, 75675600, 183783600, 1286485200, 24443218800, 38594556000, 424540116000, 733296564000, 8066262204000, 185524030692000, 1693915062840000, 5380196890068000, 38960046445320000, 166786103592108000

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

In other words, the sequence includes k iff A182860(k) > A182860(m) for all m < k.

The records for the number of distinct prime signatures are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, ... (see the link for more values). - Amiram Eldar, Jul 07 2019

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  This makes a total of 6 distinct prime signatures that appear among the unitary divisors of 60.  Since no positive integer smaller than 60 has more than 4 distinct prime signatures appearing among its unitary divisors, 60 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..60
Amiram Eldar, Table of n, a(n), A182860(a(n)) for n = 1..60
Eric Weisstein's World of Mathematics, Unitary Divisor

Subsequence of A025487, A129912, A181826, A182863. See also A034444, A085082, A182860, A182861.

f[1] = 1; f[n_] := Times @@ (Values[Counts[FactorInteger[n][[;; , 2]]]] + 1); fm = 0; s={}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Jan 19 2019 *)

a(14)-a(26) from Amiram Eldar, Jan 19 2019

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 210, 360, 420, 1260, 2310, 2520, 4620, 13860, 27720, 30030, 60060, 75600, 138600, 180180, 360360, 510510, 831600, 900900, 1021020, 1801800, 3063060, 6126120, 9699690, 10810800, 15315300, 19399380, 30630600, 37837800

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

Members m of A025487 such that A181819(m) is also a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.
Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}. - Gus Wiseman, May 21 2022

			The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.
From _Gus Wiseman_, May 21 2022: (Start)
The terms together with their sorted prime signatures and sorted prime metasignatures begin:
      1: {}                -> {}            -> {}
      2: {1}               -> {1}           -> {1}
      6: {1,2}             -> {1,1}         -> {2}
     12: {1,1,2}           -> {1,2}         -> {1,1}
     30: {1,2,3}           -> {1,1,1}       -> {3}
     60: {1,1,2,3}         -> {1,1,2}       -> {1,2}
    210: {1,2,3,4}         -> {1,1,1,1}     -> {4}
    360: {1,1,1,2,2,3}     -> {1,2,3}       -> {1,1,1}
    420: {1,1,2,3,4}       -> {1,1,1,2}     -> {1,3}
   1260: {1,1,2,2,3,4}     -> {1,1,2,2}     -> {2,2}
   2310: {1,2,3,4,5}       -> {1,1,1,1,1}   -> {5}
   2520: {1,1,1,2,2,3,4}   -> {1,1,2,3}     -> {1,1,2}
   4620: {1,1,2,3,4,5}     -> {1,1,1,1,2}   -> {1,4}
  13860: {1,1,2,2,3,4,5}   -> {1,1,1,2,2}   -> {2,3}
  27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3}   -> {1,1,3}
  30030: {1,2,3,4,5,6}     -> {1,1,1,1,1,1} -> {6}
  60060: {1,1,2,3,4,5,6}   -> {1,1,1,1,1,2} -> {1,5}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1444 terms from Amiram Eldar)
Eric Weisstein's World of Mathematics, Conjugate Partition.

Intersection of A025487 and A179983.
Subsequence of A129912 and A181826.
Includes all members of A182862.
Positions of first appearances in A353742, unordered version A238747.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A182850 gives frequency depth of prime indices, counted by A225485.
A323014 gives adjusted frequency depth of prime indices, counted by A325280.
Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

nn=1000;
r=Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,nn}];
Select[Range[nn],!MemberQ[Take[r,#-1],r[[#]]]&] (* Gus Wiseman, May 21 2022 *)

cp's OEIS Frontend

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

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

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A212167 Numbers k such that the maximum exponent in its prime factorization is not greater than the number of positive exponents (A051903(k) <= A001221(k)).

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A181824 Members of A025487 such that A025487(n) <= A181822(n).

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A181825 Members of A025487 whose prime signature is self-conjugate (as a partition).

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A181823 Members of A025487 such that A025487(n) < A181822(n).

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A181827 Members of A025487 such that A025487(n) > A181822(n).

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A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.