A181821 a(n) = smallest integer with factorization as Product p(i)^e(i) such that Product p(e(i)) = n.

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000

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

Crossrefs

Other rearrangements of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181822.

Cf. A046523, A181819, A181820.

Cf. A000040, A001221, A001222, A002110, A056239, A071625, A112798, A118914, A122111, A124859, A182850, A305936, A361808.

Sequence in context: A131117 A332290 A332293 * A241882 A238626 A168657

Adjacent sequences: A181818 A181819 A181820 * A181822 A181823 A181824

Keyword

nonn

Author

Matthew Vandermast, Dec 07 2010

Extensions

Definition corrected by Gus Wiseman, Jan 02 2019

Status

approved