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Smallest number in whose divisors the longest run is of length n, and 0 if none exists.
6

%I #7 Oct 18 2019 16:45:09

%S 0,1,2,6,12,0,60,420,840,0,2520,0,27720,0,0,360360,720720,0,12252240,

%T 0,0,0,232792560,0,5354228880,0,26771144400,0,80313433200,0,

%U 2329089562800,72201776446800,0,0,0,0,144403552893600,0,0,0,5342931457063200,0

%N Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Run_(cards)">Run (cards)</a>

%F a(n) = LCM(1,2,...,n) = A003418(n) if n + 1 is a prime power, otherwise a(n) = 0.

%t tav=Table[Max@@Length/@Split[Divisors[n],#2==#1+1&],{n,10000}];

%t Table[If[FreeQ[tav,i],0,Position[tav,i][[1,1]]],{i,0,Max@@tav}]

%Y Positions of 0's are 0 followed by A024619 - 1.

%Y The version that looks only at all divisors > 1 is A328448.

%Y The longest run of divisors of n has length A055874.

%Y The longest run of divisors of n greater than one has length A328457.

%Y Numbers whose divisors have no non-singleton runs are A005408.

%Y The number of successive pairs of divisors of n is A129308(n).

%Y The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).

%Y The smallest number whose divisors have a (not necessarily longest) maximal run of length n is A181063.

%Y Cf. A000005, A003601, A027750, A033676, A060680, A060681, A199970, A328195.

%K nonn

%O 0,3

%A _Gus Wiseman_, Oct 16 2019