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Numbers k such that the squarefree kernel of 9^k*(9^k - 1) is 3*(9^k - 1)/4.
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%I #20 Jun 15 2024 19:37:36

%S 1,3,7,9,11,13,17,19,23,27,29,31,33,37,41,43,47,49,51,53,57,59,61,67,

%T 69,71,73,77,79,81,83,87,89,91,93,97,99,101,103,107,109,111,113,119,

%U 121,123,127,129,131,133,137,139,141,143,149,151,153,157,159,161

%N Numbers k such that the squarefree kernel of 9^k*(9^k - 1) is 3*(9^k - 1)/4.

%C From _Artur Jasinski_: (Start)

%C The maximal value of the squarefree kernel of a*b*9^k for every number 9^k and every a,b such that a + b = 9^k and gcd(a,b,3)=1 is never less than 3*(9^k - 1)/4 and is exactly equal to 3*(9^k - 1)/4 for exponents k in this sequence.

%C Conjecture: This sequence is infinite. (End)

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

%o isok(k) = rad(9^k*(9^k - 1)) == 3*(9^k - 1)/4; \\ _Michel Marcus_, Dec 24 2022

%Y Cf. A007947, A054880

%K nonn,hard

%O 1,2

%A _Artur Jasinski_, Jan 29 2010

%E Edited by _Jon E. Schoenfield_, Dec 23 2022

%E More terms from _Sean A. Irvine_, Jun 15 2024