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A324385 Distance from the n-th highly composite number, A002182(n), from the largest prime <= A002182(n).

%I A324385 #28 Dec 25 2020 10:51:56
%S A324385 0,1,1,1,1,5,1,1,7,1,1,1,1,1,1,11,17,1,1,1,13,11,11,19,17,13,1,23,1,1,
%T A324385 13,17,17,13,17,1,17,1,1,23,17,17,17,1,19,83,37,23,17,23,1,43,19,1,19,
%U A324385 43,19,31,23,19,31,19,19,1,1,1,1,47,1,31,47,23,53,23,83,37,31,1,31,1,23,61,1,41,47,61,41,29,41,29,43,73,29,47,31,31
%N A324385 Distance from the n-th highly composite number, A002182(n), from the largest prime <= A002182(n).
%C A324385 Like in A141345 it appears (or is conjectured) that no composite numbers ever occur here. Taken together, this leads to McEachen's conjecture given in A117825. Here in range 2..10000 term 1 occurs for 313 times.
%C A324385 The arithmetic mean of a(n)/log(A002182(n)) for the terms 3..10000 is 1.513, i.e., a rough approximation is given by a(n) ~ log(A002182(n)^(3/2)). - _A.H.M. Smeets_, Dec 02 2020
%H A324385 Antti Karttunen, <a href="/A324385/b324385.txt">Table of n, a(n) for n = 2..10000</a>
%F A324385 a(n) = A002182(n) - A007917(A002182(n)).
%e A324385 A002182(2) = 2, the largest prime <= 2 is 2 itself, thus a(2) = 2-2 = 0.
%e A324385 A002182(7) = 36, the largest prime <= 36 is 31, thus a(7) = 36-31 = 5.
%t A324385 With[{s = Array[DivisorSigma[0, #] &, 10^6]}, {0}~Join~Map[# - NextPrime[#, -1] &@ FirstPosition[s, #][[1]] &, Drop[Union@ FoldList[Max, s], 2]]] (* or *)
%t A324385 {0}~Join~Map[# - NextPrime[#, -1] &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[3 ;; 97, -1]] ] (* _Michael De Vlieger_, Dec 11 2020 *)
%o A324385 (PARI) A324385(n) = (A002182(n)-precprime(A002182(n)));
%Y A324385 Cf. A002182, A007917, A060270, A117825, A141345.
%K A324385 nonn
%O A324385 2,6
%A A324385 _Antti Karttunen_, Feb 26 2019