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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Examples

			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.
		

Links

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

Crossrefs

A025487 gives range of values of this sequence.

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

Cf. A181819, A181821.

Programs

Haskell

import Data.List (sort)
a046523 = product .
          zipWith (^) a000040_list . reverse . sort . a124010_row
-- Reinhard Zumkeller, Apr 27 2013

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 *)

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

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

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

Extensions

Corrected and extended by Ray Chandler, Mar 11 2004