A046523 Smallest number with same prime signature as n.

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, 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, 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

Offset

1,2

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Formula

In prime factorization of n, replace most common prime by 2, next most common by 3, etc.

a(n) = A124859(A124859(n)) = A181822(A124859(n)). - Matthew Vandermast, May 19 2012

a(n) = A181821(A181819(n)). - Alois P. Heinz, Feb 17 2020

Example

If p,q,... are different primes, a(p)=2, a(p^2)=4, a(pq)=6, a(p^2*q)=12, etc.
n = 108 = 2*2*3*3*3 is replaced by a(n) = 2*2*2*3*3 = 72;
n = 105875 = 5*5*5*7*11*11 is represented by a(n) = 2*2*2*3*3*5 = 360.
Prime-powers are replaced by corresponding powers of 2, primes by 2.
Factorials, primorials and lcm[1..n] are in the sequence.
A000005(a(n)) = A000005(n) remains invariant; least and largest prime factors of a(n) are 2 or p[A001221(n)] resp.

Maple

a:= n-> (l-> mul(ithprime(i)^l[i][2], i=1..nops(l)))
        (sort(ifactors(n)[2], (x, y)->x[2]>y[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 18 2014

Mathematica

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]]
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 *)

Prog

(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
(PARI) A046523(n)=factorback(primes(#n=vecsort(factor(n)[, 2], , 4)), n) \\ M. F. Hasler, Oct 12 2018, improved Jul 18 2019
(Haskell)
import Data.List (sort)
a046523 = product .
          zipWith (^) a000040_list . reverse . sort . a124010_row
-- Reinhard Zumkeller, Apr 27 2013
(Python)
from sympy import factorint
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1 # Indranil Ghosh, May 05 2017
(Python)
from math import prod
from sympy import factorint, prime
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

Crossrefs

A025487 gives range of values of this sequence.

Cf. A000142, A002110, A003418, A001221, A000040, A000005, A124010, A071364, A085079, A089247.

Cf. A181819, A181821.

Sequence in context: A083260 A284011 A275468 * A278524 A278523 A071364

Adjacent sequences: A046520 A046521 A046522 * A046524 A046525 A046526

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Corrected and extended by Ray Chandler, Mar 11 2004

Status

approved