A396513 - OEIS

%I #17 Jun 02 2026 08:44:17

%S 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,

%T 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,

%U 106,36,11,16,38,116,118,5,122,4,2,128,130,12,134,136,6,20,142,18,146

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

%C 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.

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

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

%H Banri Ogawa, <a href="/A396513/b396513.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

%o (Python)

%o from math import prod

%o from sympy import primefactors

%o def a(n): return next(k for k in range(1, 2*n+1) if prod(primefactors(k)) == prod(primefactors(k+2*n)))

%o (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947

%o a(n) = my(k=1); while(rad(k+2*n) != rad(k), k++); k; \\ _Michel Marcus_, May 28 2026

%Y Cf. A007947, A396510 (maximum k).

%K nonn

%O 1,1

%A _Banri Ogawa_, May 28 2026