A396513 a(n) is the minimum integer k such that rad(k)=rad(k+2n), where rad(x) is the squarefree kernel of x (A007947).

2, 4, 2, 8, 10, 4, 2, 16, 6, 5, 22, 3, 26, 4, 2, 32, 34, 12, 38, 10, 6, 44, 46, 6, 50, 52, 18, 8, 58, 4, 2, 64, 6, 68, 10, 9, 74, 76, 3, 20, 82, 12, 86, 88, 6, 92, 94, 12, 14, 25, 6, 104, 106, 36, 11, 16, 38, 116, 118, 5, 122, 4, 2, 128, 130, 12, 134, 136, 6, 20, 142, 18, 146

Offset

1,1

Comments

Note that the equation rad(k)=rad(k+n) has no solutions for odd n. Thus, parameterizing the difference as 2n is the natural formulation.

The existence of at least one solution for any n is guaranteed because k=2n always satisfies the condition. Thus, a(n)<=2n.

The prime factors of k must be a subset of the prime factors of 2n.

Links

Banri Ogawa, Table of n, a(n) for n = 1..10000

Example

For n=10, rad(k)=rad(k+20). The minimum k satisfying this is a(10)=5 (rad(5)=rad(5+20)=5).

Mathematica

a[n_]:=Module[{k=1}, Until[Times@@First/@FactorInteger@k==Times@@First/@FactorInteger@(k+2n), k++]; k]; Array[a, 73] (* James C. McMahon, Jun 01 2026 *)

Prog

(Python)
from math import prod
from sympy import primefactors
def a(n): return next(k for k in range(1, 2*n+1) if prod(primefactors(k)) == prod(primefactors(k+2*n)))
(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = my(k=1); while(rad(k+2*n) != rad(k), k++); k; \\ Michel Marcus, May 28 2026

Crossrefs

Cf. A007947, A396510 (maximum k).

Sequence in context: A324716 A328378 A296429 * A065286 A068217 A303603

Adjacent sequences: A396510 A396511 A396512 * A396514 A396515 A396516

Keyword

nonn

Author

Banri Ogawa, May 28 2026

Status

approved