A385644 Swap multiplication and exponentiation in the canonical prime factorization of n.

2, 3, 4, 5, 8, 7, 6, 6, 32, 11, 64, 13, 128, 243, 8, 17, 64, 19, 1024, 2187, 2048, 23, 216, 10, 8192, 9, 16384, 29, 14134776518227074636666380005943348126619871175004951664972849610340958208, 31, 10, 177147, 131072, 78125, 4096, 37, 524288, 1594323, 7776, 41

Offset: 2

Author

Jens Ahlström, Jul 06 2025

Keywords

• nonn

Comments

In the canonical prime factorization of n larger than one, swap multiplication and exponentiation and calculate the result.

Examples

			a(6) = a(2 * 3) = 2^3 = 8,
a(24) = a(2^3 * 3) = (2 * 3)^3 = 216,
a(30) = a(2 * 3 * 5) = 2^3^5 = 2^243.
		

Links

• Jens Ahlström, Table of n, a(n) for n = 2..65

Crossrefs

Cf. A001414, A000026, A005361, A008474.

Programs

Mathematica

f[{p_,e_}]:=p*e;a[n_]:=Module[{pp=f/@FactorInteger[n]},r=pp[[-1]];Do[r=pp[[Length[pp]-i]]^r,{i,1,Length[pp]-1}];r];Array[a,40,2] (* James C. McMahon, Jul 11 2025 *)
A385644[n_] := Power @@ Times @@@ FactorInteger[n];
Array[A385644, 40, 2] (* Paolo Xausa, Jul 14 2025 *)

Python

from sympy import factorint
from functools import reduce
def rpow(a, b):
    return b**a
def a(n):
    return reduce(rpow, [p*e for p, e in reversed(factorint(n).items())])
print([a(n) for n in range(2, 42)])