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Maximal digit in the primorial base representation of the n-th arithmetic derivative of 128.
1

%I #11 Feb 05 2024 18:09:11

%S 4,4,6,2,6,11,8,7,11,11,8,10,15,15,12,18,17,30,28,22,21,37,28,38,42,

%T 33,22,52,43,56,47,49,57,60,55,54,58,70,52,66,68,57,63,86,58,88,92,66,

%U 78,95,85,52,102,70,111,57,117,99,136,104,129,110,146,127,135,132,131,129,126,145,112,150,128,129,154,161,145

%N Maximal digit in the primorial base representation of the n-th arithmetic derivative of 128.

%C This sequence relates to the question whether A327969(128) has a positive integer value, or whether it is -1 by the escape clause.

%C Note that when iterating the map k -> k' from A276086(A369638(4)) = A276086(15232) = 3299611946113357875 onward, the maximal exponent in the prime factorization (A051903) keeps on decreasing until it reaches 1 at the fifth iteration, and then stays as 1 for three more iterations (with k then 38863666759992439 = 643*60441161368573), but then, alas, on the next iteration, k' = 60441161369216 = 2^7 * 472196573197.

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F a(n) = A328114(A369638(n)).

%e The third arithmetic derivative (A099306) of 128 is 5056, which in primorial base (A049345) is written as 220220, therefore a(3) = 2.

%e The fourth arithmetic derivative (A258644) of 128 is 15232, which in primorial base is written as 663320, therefore a(4) = 6.

%Y Cf. A003415, A049345, A099306, A258644, A328114, A369638.

%Y Cf. also A327969.

%K nonn

%O 0,1

%A _Antti Karttunen_, Feb 04 2024