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

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

1,2

Comments

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).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Conjugate Partition

Formula

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)).

Example

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.

Mathematica

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 *)

Prog

(PARI) 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
(Python)
from math import prod
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log, primorial, factorint
from oeis_sequences.OEISsequences import bisection
def A181822(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 prod(primorial(e) for e in factorint(bisection(f, n, n)).values()) # Chai Wah Wu, Mar 23 2026

Crossrefs

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.

Sequence in context: A064538 A002790 A108951 * A346107 A174940 A293011

Adjacent sequences: A181819 A181820 A181821 * A181823 A181824 A181825

Keyword

nonn,look

Author

Matthew Vandermast, Dec 07 2010

Status

approved