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
EXAMPLE
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
KEYWORD
nonn
AUTHOR
Matthew Vandermast, Dec 07 2010
EXTENSIONS
Definition corrected by Gus Wiseman, Jan 02 2019
STATUS
approved