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A117825 Distance from n-th highly composite number (cf. A002182) to nearest prime.

%I A117825 #49 Aug 10 2020 09:33:37
%S A117825 1,0,1,1,1,1,1,1,1,7,1,1,1,1,1,1,11,1,1,1,1,1,1,11,13,1,11,1,17,1,1,
%T A117825 13,13,1,1,17,1,17,1,1,17,17,17,1,1,19,37,37,1,17,23,1,29,1,1,19,1,19,
%U A117825 23,1,19,31,1,19,1,1,1,1,23,1,29,23,23,1,23,71,37,1,1,31,1,23,53,1,31
%N A117825 Distance from n-th highly composite number (cf. A002182) to nearest prime.
%C A117825 a) Conjecture: entries > 1 will always be prime. The entry will be larger than the largest prime factor of the highly composite number.
%C A117825 b) Will 1 always be the most common entry?
%C A117825 c) While a prime may always be located close to each highly composite number, is the converse false?
%C A117825 d) Is there always a prime between successive highly composite numbers?
%C A117825 From _Antti Karttunen_, Feb 26 2019: (Start)
%C A117825 The second sentence of point (a) follows as both gcd(n, A151799(n)) = 1 and gcd(A151800(n), n) = 1 for all n > 2 and the fact that the highly composite numbers are products of primorials, A002110 (with the least coprime prime > the largest prime factor). See also the conjectures and notes in A129912 and A141345. (End)
%H A117825 Charles R Greathouse IV, <a href="/A117825/b117825.txt">Table of n, a(n) for n = 1..19999</a>
%H A117825 Bill McEachen, <a href="https://commons.wikimedia.org/wiki/File:A117825.svg">Alternate plot</a>, Wikimedia Commons.
%H A117825 Graeme McRae, <a href="https://web.archive.org/web/20190223125015/http://2000clicks.com/mathhelp/NumberFactorsHighlyComposite.aspx">Highly Composite Numbers</a>.
%H A117825 Wikipedia, <a href="http://en.wikipedia.org/wiki/Highly_composite">Highly Composite Numbers</a>.
%H A117825 Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisor_function">Divisor Function (sigma)</a>.
%F A117825 a(1) = 1; for n > 1, a(n) = min(A141345(n), A324385(n)). - _Antti Karttunen_, Feb 26 2019
%e A117825 a(5) = abs(12-11) = 1.
%t A117825 With[{s = DivisorSigma[0, Range[Product[Prime@ i, {i, 8}]]]}, Map[If[PrimeQ@ #, 0, Min[# - NextPrime[#, -1], NextPrime[#] - #]] &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* _Michael De Vlieger_, Mar 11 2019 *)
%o A117825 (PARI)
%o A117825 A141345(n) = (nextprime(1+A002182(n))-A002182(n));
%o A117825 A324385(n) = (A002182(n)-precprime(A002182(n)));
%o A117825 A117825(n) = if(1==n,1,min(A141345(n), A324385(n))); \\ _Antti Karttunen_, Feb 26 2019
%Y A117825 Cf. A002100, A002182, A007917, A129912, A141345, A151800, A324385.
%Y A117825 Sequences tied to conjecture a): A228943, A228945.
%Y A117825 Cf. also A005235, A060270.
%K A117825 nonn,look
%O A117825 1,10
%A A117825 _Bill McEachen_, May 01 2006
%E A117825 More terms from _Don Reble_, May 02 2006