%I #61 Feb 04 2022 14:26:35
%S 1,2,2,4,2,6,2,8,4,6,2,12,2,6,6,16,2,12,2,12,6,6,2,24,4,6,8,12,2,30,2,
%T 32,6,6,6,36,2,6,6,24,2,30,2,12,12,6,2,48,4,12,6,12,2,24,6,24,6,6,2,
%U 60,2,6,12,64,6,30,2,12,6,30,2,72,2,6,12,12,6,30,2,48,16,6,2,60,6,6,6,24,2
%N Smallest number with same prime signature as n.
%H T. D. Noe, <a href="/A046523/b046523.txt">Table of n, a(n) for n = 1..10000</a>
%F In prime factorization of n, replace most common prime by 2, next most common by 3, etc.
%F a(n) = A124859(A124859(n)) = A181822(A124859(n)). - _Matthew Vandermast_, May 19 2012
%F a(n) = A181821(A181819(n)). - _Alois P. Heinz_, Feb 17 2020
%e If p,q,... are different primes, a(p)=2, a(p^2)=4, a(pq)=6, a(p^2*q)=12, etc.
%e n = 108 = 2*2*3*3*3 is replaced by a(n) = 2*2*2*3*3 = 72;
%e n = 105875 = 5*5*5*7*11*11 is represented by a(n) = 2*2*2*3*3*5 = 360.
%e Prime-powers are replaced by corresponding powers of 2, primes by 2.
%e Factorials, primorials and lcm[1..n] are in the sequence.
%e A000005(a(n)) = A000005(n) remains invariant; least and largest prime factors of a(n) are 2 or p[A001221(n)] resp.
%p a:= n-> (l-> mul(ithprime(i)^l[i][2], i=1..nops(l)))
%p (sort(ifactors(n)[2], (x, y)->x[2]>y[2])):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Aug 18 2014
%t Table[Apply[Times, p[w]^Reverse[Sort[ex[w]]]], {w, 1, 1000}] p[x_] := Table[Prime[w], {w, 1, lf[x]}] ex[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]]
%t ps[n_] := Sort[Last /@ FactorInteger[n]]; Join[{1}, Table[i = 2; While[ps[n] != ps[i], i++]; i, {n, 2, 89}]] (* _Jayanta Basu_, Jun 27 2013 *)
%o (PARI) a(n)=my(f=vecsort(factor(n)[,2],,4),p);prod(i=1,#f,(p=nextprime(p+1))^f[i]) \\ _Charles R Greathouse IV_, Aug 17 2011
%o (PARI) A046523(n)=factorback(primes(#n=vecsort(factor(n)[,2],,4)),n) \\ _M. F. Hasler_, Oct 12 2018, improved Jul 18 2019
%o (Haskell)
%o import Data.List (sort)
%o a046523 = product .
%o zipWith (^) a000040_list . reverse . sort . a124010_row
%o -- _Reinhard Zumkeller_, Apr 27 2013
%o (Python)
%o from sympy import factorint
%o def P(n):
%o f = factorint(n)
%o return sorted([f[i] for i in f])
%o def a(n):
%o x=1
%o while True:
%o if P(n) == P(x): return x
%o else: x+=1 # _Indranil Ghosh_, May 05 2017
%o (Python)
%o from math import prod
%o from sympy import factorint, prime
%o def A046523(n): return prod(prime(i+1)**e for i,e in enumerate(sorted(factorint(n).values(),reverse=True))) # _Chai Wah Wu_, Feb 04 2022
%Y A025487 gives range of values of this sequence.
%Y Cf. A000142, A002110, A003418, A001221, A000040, A000005, A124010, A071364, A085079, A089247.
%Y Cf. A181819, A181821.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E Corrected and extended by _Ray Chandler_, Mar 11 2004