cp's OEIS Frontend

A335407 Number of anti-run permutations of the prime indices of n!.

%I A335407 #18 Feb 03 2021 15:55:53
%S A335407 1,1,1,2,0,2,3,54,0,30,105,6090,1512,133056,816480,127209600,0,
%T A335407 10090080,562161600,69864795000,49989139200,29593652088000,
%U A335407 382147120555200,41810689605484800,4359985823793600,3025062801079038720,49052072750637116160,25835971971637227375360
%N A335407 Number of anti-run permutations of the prime indices of n!.
%C A335407 An anti-run is a sequence with no adjacent equal parts.
%C A335407 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A335407 Conjecture: Only vanishes at n = 4 and n = 8.
%C A335407 a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - _David A. Corneth_, Jul 04 2020
%H A335407 Andrew Howroyd, <a href="/A335407/b335407.txt">Table of n, a(n) for n = 0..200</a>
%F A335407 a(n) = A335452(A000142(n)). - _Andrew Howroyd_, Feb 03 2021
%e A335407 The a(0) = 1 through a(6) = 3 anti-run permutations:
%e A335407   ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
%e A335407                (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
%e A335407                                       (1,3,1,2,1,2,1)
%t A335407 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335407 Table[Length[Select[Permutations[primeMS[n!]],!MatchQ[#,{___,x_,x_,___}]&]],{n,0,10}]
%o A335407 (PARI) \\ See A335452 for count.
%o A335407 a(n)={count(factor(n!)[,2])} \\ _Andrew Howroyd_, Feb 03 2021
%Y A335407 The version for Mersenne numbers is A335432.
%Y A335407 Anti-run compositions are A003242.
%Y A335407 Anti-run patterns are counted by A005649.
%Y A335407 Permutations of prime indices are A008480.
%Y A335407 Anti-runs are ranked by A333489.
%Y A335407 Separable partitions are ranked by A335433.
%Y A335407 Inseparable partitions are ranked by A335448.
%Y A335407 Anti-run permutations of prime indices are A335452.
%Y A335407 Strict permutations of prime indices are A335489.
%Y A335407 Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
%Y A335407 Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.
%K A335407 nonn
%O A335407 0,4
%A A335407 _Gus Wiseman_, Jul 01 2020
%E A335407 Terms a(14) and beyond from _Andrew Howroyd_, Feb 03 2021