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A046523 Smallest number with same prime signature as n. 101
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 (list; graph; refs; listen; history; text; internal format)
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

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

(Haskell)

import Data.List (sort)

a046523 = product .

          zipWith (^) a000040_list . reverse . sort . a124010_row

-- Reinhard Zumkeller, Apr 27 2013

CROSSREFS

A025487 gives range of values of this sequence.

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

Sequence in context: A119655 A083260 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

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Last modified December 10 03:42 EST 2016. Contains 278993 sequences.