%I #42 Jul 21 2020 02:30:26

%S 2,55,171,183,203

%N Numbers m such that 5^m - 4^m is not squarefree, but 5^d - 4^d is squarefree for every proper divisor d of m.

%C Where numbers m such that 5^m - 4^m is not squarefree: numbers of the form i*a(j) for i >= 1.

%C 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:

%C A237043 (k = 1): 6, 20, 21, 110, 136, 155, 253, 364, 602, 657, 812, 889, 979, 1081, ... a(15) >= 1207 - _Max Alekseyev_, Sep 28 2015;

%C A280203 (k = 2): 10, 11, 42, 52, 57, 203, 272, 497, ... a(9) > 497 - _Charles R Greathouse IV_, Dec 27 2016;

%C A280208 (k = 3): 4, 14, 55, 78, 111, 253, 342, 355, ... a(9) >= 431;

%C this sequence (k = 4): 2, 55, 171, 183, 203, ... a(6) >= 367;

%C A... (k = 5): 21, 22, 39, 136, 186, 203, 244, 333, ... a(9) >= 337;

%C A280307 (k = 6): 20, 26, 55, 68, 171, 258, 310, ... a(8) >= 323;

%C A... (k = 7): 3, 10, 55, 272, ... a(5) >= 289;

%C A... (k = 8): 20, 21, 22, 34, 93, 116, 138, 156, 166, 205, 253, ... a(12) >= 277;

%C A... (k = 9): 33, 38, 42, 78, 110, 155, ... a(7) >= 263;

%C A... (k = 10): 6, 14, 52, 68, 253, ... a(6) >= 263;

%C A... (k = 11): 20, 42, 46, 53, 114, 136, 156, 205, ... a(9) >= 251.

%C The smallest square of 5^m - 4^m as defined above are 9, 121, 361, 3721, 841. - _Robert Price_, Mar 07 2017

%C a(6) <= 465. 465, 955, 1027 are terms. - _Chai Wah Wu_, Jul 20 2020

%e 2 is in this sequence because 5^1 - 4^1 = 1 is squarefree where 1 is proper divisor of 2 and 5^2 - 4^2 = 9 = 3^2 is not squarefree.

%Y Cf. A005060, A237043, A280203, A280208.

%K nonn,more

%O 1,1

%A _Juri-Stepan Gerasimov_, Dec 28 2016

%E a(3)-a(5) from _Jinyuan Wang_, May 15 2020