%I #21 Feb 21 2023 07:31:49

%S 1,2,6,4,30,12,210,60,8,2310,36,420,24,30030,180,4620,120,510510,1260,

%T 72,60060,16,900,840,9699690,13860,360,1021020,48,6300,9240,223092870,

%U 180180,2520,19399380,240,69300,216,120120,6469693230,1800,3063060,144,44100,27720,446185740,1680,900900,1080,2042040,200560490130,12600,58198140,32,720

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

%C A permutation of the members of A025487.

%C 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).

%H Charles R Greathouse IV, <a href="/A181822/b181822.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConjugatePartition.html">Conjugate Partition</a>

%F 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)).

%e 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.

%t 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 *)

%o (PARI) partitionConj(v)=vector(v[1],i,sum(j=1,#v,v[j]>=i))

%o primeSignature(n)=vecsort(factor(n)[,2]~,,4)

%o f(n)=if(n==1, return(1)); my(e=partitionConj(primeSignature(n))~); factorback(concat(Mat(primes(#e)~),e))

%o 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];

%o concat(1, apply(f, A025487)) \\ _Charles R Greathouse IV_, Jun 02 2016

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

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

%K nonn,look

%O 1,2

%A _Matthew Vandermast_, Dec 07 2010