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Distance from the n-th highly composite number, A002182(n), to the next prime.
6

%I #19 Dec 25 2020 10:51:38

%S 1,1,1,1,1,5,1,5,1,7,1,1,7,7,13,17,13,1,11,1,11,1,1,19,13,1,11,1,17,1,

%T 29,13,13,1,1,17,13,23,17,19,17,17,19,1,19,23,37,53,1,17,29,43,29,1,

%U 19,19,1,23,23,1,41,41,1,53,29,19,19,23,23,47,29,23,37,1,59,71,41,1,29,37

%N Distance from the n-th highly composite number, A002182(n), to the next prime.

%C It appears that (1) every term is either 1 or a prime and (2) every prime greater than 3 appears. Note that a prime can occur only a finite number of times. Similar to Fortune's conjecture (A005235) and McEachen's conjecture (A117825).

%C 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 T. D. Noe, <a href="/A141345/b141345.txt">Table of n, a(n) for n=1..10000</a>

%t With[{s = Array[DivisorSigma[0, #] &, 10^6]}, Map[NextPrime[#] - # &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* or *)

%t Map[NextPrime[#] - # &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[1 ;; 80, -1]] ] (* _Michael De Vlieger_, Dec 11 2020 *)

%K nonn

%O 1,6

%A _T. D. Noe_, Jun 26 2008