A280208 - OEIS

%I #31 Jul 21 2020 02:30:15

%S 4,14,55,78,111,253,342,355

%N Numbers m such that 4^m - 3^m is not squarefree, but 4^d - 3^d is squarefree for every proper divisor d of m.

%C Where numbers m such that 4^m - 3^m is not squarefree: numbers of the form i*a(j) for i >= 1.

%C The smallest squares of 4^m - 3^m as defined above are 25, 49, 121, 169, 1369, 529, 361, 5041. - _Robert Price_, Mar 07 2017

%C 431 <= a(9) <= 1081. 1081, 3403 are terms. - _Chai Wah Wu_, Jul 20 2020

%e 4 is in this sequence because all 4^1 - 3^1 = 1, 4^2 - 3^2 = 7 are squarefrees where 1, 2 are proper divisors of 4 and 4^4 - 3^4 = 175 = 7*5^2 is not squarefree;

%e 14 is in this sequence because all 4^1 = 3^2 = 1, 4^2 - 3^2 = 7, 4^7 - 3^7 = 14197 are squarefrees where 1, 2, 7 are proper divisors of 14 and 4^14 - 3^14 = 263652487 = 7^2*3591*14197 is not squarefree.

%t Function[s, DeleteCases[#, 0] &@ MapIndexed[#1 Boole[! AnyTrue[Take[s, First@ #2 - 1], Function[k, Divisible[#1, k]]]] &, s]]@ Select[Range@ 80, ! SquareFreeQ[4^# - 3^#] &] (* _Michael De Vlieger_, Dec 30 2016 *)

%Y Cf. A005061.

%Y Cf. Numbers m such that (k+1)^m - k^m is not squarefree, but (k+1)^d - k^d is squarefree for every proper divisor d of m: A237043 (k = 1), A280203 (k = 2), this sequence (k = 3), A280209 (k = 4), A280307 (k = 6).

%K nonn,more

%O 1,1

%A _Juri-Stepan Gerasimov_, Dec 28 2016

%E a(6)-a(8) from _Jinyuan Wang_, May 15 2020