A304768 Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 18, 13, 14, 15, 2, 17, 12, 19, 50, 21, 22, 23, 54, 5, 26, 3, 98, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 7, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 2, 65, 66, 67

Offset: 1

Author

Gus Wiseman, May 18 2018

Keywords

• nonn

Comments

Image is the weak numbers A052485, on which n -> a(n) is an involution whose fixed points are the squarefree numbers A005117.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597, A001694, A005117, A007916, A007947, A013929, A051903, A052410, A062759, A066638, A072774, A087320, A303554.

Programs

Mathematica

acj[n_]:=Module[{f,m},f=FactorInteger[n];m=Max[Last/@f];Times@@Table[p[[1]]^(m-p[[2]]+1),{p,f}]];
Array[acj,100]

PARI

a(n) = {if(n==1, 1, my(f = factor(n), e = vecmax(f[,2]) + 1); prod(i = 1, #f~, f[i,1]^e) / n);} \\ Amiram Eldar, Feb 12 2023

Formula

If n = Product_{i = 1..k} prime(x_i)^y_i, then a(n) = Product_{i = 1..k} prime(x_i)^(max{y_1,...,y_k} - y_i + 1).