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A181821
a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.
122
1, 2, 4, 6, 8, 12, 16, 30, 36, 24, 32, 60, 64, 48, 72, 210, 128, 180, 256, 120, 144, 96, 512, 420, 216, 192, 900, 240, 1024, 360, 2048, 2310, 288, 384, 432, 1260, 4096, 768, 576, 840, 8192, 720, 16384, 480, 1800, 1536, 32768, 4620, 1296, 1080, 1152, 960, 65536
OFFSET
1,2
COMMENTS
A permutation of A025487. a(n) is the member m of A025487 such that A181819(m) = n. a(n) is also the member of A025487 whose prime signature is conjugate to the prime signature of A108951(n).
If n = Product_i prime(e(i)) with the e(i) weakly decreasing, then a(n) = Product_i prime(i)^e(i). For example, 90 = prime(3) * prime(2) * prime(2) * prime(1), so a(90) = prime(1)^3 * prime(2)^2 * prime(3)^2 * prime(4)^1 = 12600. - Gus Wiseman, Jan 02 2019
LINKS
Eric Weisstein's World of Mathematics, Conjugate Partition
FORMULA
If A108951(n) = Product p(i)^e(i), then a(n) = Product A002110(e(i)). I.e., a(n) = A108951(A181819(A108951(n))).
a(A181819(n)) = A046523(n). - [See also A124859]. Antti Karttunen, Dec 10 2018
a(n) = A025487(A361808(n)). - Pontus von Brömssen, Mar 25 2023
a(n) = A108951(A122111(n)). - Antti Karttunen, Sep 15 2023
EXAMPLE
The canonical factorization of 24 is 2^3*3^1. Therefore, p(e(i)) = prime(3)*prime(1)(i.e., A000040(3)*A000040(1)), which equals 5*2 = 10. Since 24 is the smallest integer for which p(e(i)) = 10, a(10) = 24.
MAPLE
a:= n-> (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(sort(map(i->
numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..70); # Alois P. Heinz, Sep 05 2018
MATHEMATICA
With[{s = Array[If[# == 1, 1, Times @@ Map[Prime@ Last@ # &, FactorInteger@ #]] &, 2^16]}, Array[First@ FirstPosition[s, #] &, LengthWhile[Differences@ Union@ s, # == 1 &]]] (* Michael De Vlieger, Dec 17 2018 *)
Table[Times@@MapIndexed[Prime[#2[[1]]]^#1&, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], {n, 30}] (* Gus Wiseman, Jan 02 2019 *)
PROG
(PARI) A181821(n) = { my(f=factor(n), p=0, m=1); forstep(i=#f~, 1, -1, while(f[i, 2], f[i, 2]--; m *= (p=nextprime(p+1))^primepi(f[i, 1]))); (m); }; \\ Antti Karttunen, Dec 10 2018
(Python)
from math import prod
from sympy import prime, primepi, factorint
def A181821(n): return prod(prime(i)**e for i, e in enumerate(sorted(map(primepi, factorint(n, multiple=True)), reverse=True), 1)) # Chai Wah Wu, Sep 15 2023
KEYWORD
nonn
AUTHOR
Matthew Vandermast, Dec 07 2010
EXTENSIONS
Definition corrected by Gus Wiseman, Jan 02 2019
STATUS
approved