%I #49 Aug 10 2020 09:33:37

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

%C a) Conjecture: entries > 1 will always be prime. The entry will be larger than the largest prime factor of the highly composite number.

%C b) Will 1 always be the most common entry?

%C c) While a prime may always be located close to each highly composite number, is the converse false?

%C d) Is there always a prime between successive highly composite numbers?

%C From _Antti Karttunen_, Feb 26 2019: (Start)

%C 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 Charles R Greathouse IV, <a href="/A117825/b117825.txt">Table of n, a(n) for n = 1..19999</a>

%H Bill McEachen, <a href="https://commons.wikimedia.org/wiki/File:A117825.svg">Alternate plot</a>, Wikimedia Commons.

%H Graeme McRae, <a href="https://web.archive.org/web/20190223125015/http://2000clicks.com/mathhelp/NumberFactorsHighlyComposite.aspx">Highly Composite Numbers</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Highly_composite">Highly Composite Numbers</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisor_function">Divisor Function (sigma)</a>.

%F a(1) = 1; for n > 1, a(n) = min(A141345(n), A324385(n)). - _Antti Karttunen_, Feb 26 2019

%e a(5) = abs(12-11) = 1.

%t 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 (PARI)

%o A141345(n) = (nextprime(1+A002182(n))-A002182(n));

%o A324385(n) = (A002182(n)-precprime(A002182(n)));

%o A117825(n) = if(1==n,1,min(A141345(n), A324385(n))); \\ _Antti Karttunen_, Feb 26 2019

%Y Cf. A002100, A002182, A007917, A129912, A141345, A151800, A324385.

%Y Sequences tied to conjecture a): A228943, A228945.

%Y Cf. also A005235, A060270.

%K nonn,look

%O 1,10

%A _Bill McEachen_, May 01 2006

%E More terms from _Don Reble_, May 02 2006