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A342018 Numbers k such that the arithmetic derivative of A276086(k) is divisible by at least one prime power divisor of the form p^p, where A276086 gives the prime product form of primorial base expansion of its argument.

%I A342018 #22 Mar 12 2021 23:12:04
%S A342018 8,16,24,36,44,52,64,72,80,88,92,100,108,116,120,126,128,136,144,156,
%T A342018 164,172,184,192,200,208,216,222,224,232,244,252,260,268,271,272,280,
%U A342018 288,296,300,308,316,324,336,344,348,352,364,372,380,388,392,397,400,408,416,424,432,440,444,448,452,460,468,476,480,488,493,496
%N A342018 Numbers k such that the arithmetic derivative of A276086(k) is divisible by at least one prime power divisor of the form p^p, where A276086 gives the prime product form of primorial base expansion of its argument.
%C A342018 Numbers k for which A342019(k) > 0, or equally, A342007(A327860(k)) = A342017(k) is larger than one, or equally A342007(A342002(k)) > 1, that is, k for which A342023(A342002(k)) = 1.
%C A342018 The first odd term is a(35) = 271.
%H A342018 Antti Karttunen, <a href="/A342018/b342018.txt">Table of n, a(n) for n = 1..10000</a>
%H A342018 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%e A342018 8 is present as A276086(8) = 15, A003415(15) = 8 = 2^3, which is thus divisible by p^p (with p=2 in this case).
%e A342018 271 is present as A276086(271) = 1078, A003415(1078) = 945 = 3^3 * 5 * 7, which is thus divisible by p^p (with p=3 in this case).
%t A342018 Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Position[#, _?(# > 1 &)][[All, 1]] &@ Array[Function[k, Times @@ Map[#1^Floor[#2/#1] & @@ # &, FactorInteger[#]] &@ If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[#, b] &, 500]] (* _Michael De Vlieger_, Mar 12 2021 *)
%o A342018 (PARI) isA342018(n) = (A342017(n)>1);
%Y A342018 Positions of terms larger than one in A342017, of nonzero terms in A342019.
%Y A342018 Cf. A003415, A276086, A327860, A342002, A342007, A342017, A342019, A342023.
%Y A342018 Not a subsequence of A342006.
%K A342018 nonn
%O A342018 1,1
%A A342018 _Antti Karttunen_, Mar 04 2021
%E A342018 Name changed by _Antti Karttunen_, Mar 12 2021