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A335407
Number of anti-run permutations of the prime indices of n!.
8
1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360
OFFSET
0,4
COMMENTS
An anti-run is a sequence with no adjacent equal parts.
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.
Conjecture: Only vanishes at n = 4 and n = 8.
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
LINKS
FORMULA
a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021
EXAMPLE
The a(0) = 1 through a(6) = 3 anti-run permutations:
() () (1) (1,2) . (1,2,1,3,1) (1,2,1,2,1,3,1)
(2,1) (1,3,1,2,1) (1,2,1,3,1,2,1)
(1,3,1,2,1,2,1)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Permutations[primeMS[n!]], !MatchQ[#, {___, x_, x_, ___}]&]], {n, 0, 10}]
PROG
(PARI) \\ See A335452 for count.
a(n)={count(factor(n!)[, 2])} \\ Andrew Howroyd, Feb 03 2021
CROSSREFS
The version for Mersenne numbers is A335432.
Anti-run compositions are A003242.
Anti-run patterns are counted by A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Sequence in context: A220222 A089839 A206823 * A151668 A086151 A099040
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 01 2020
EXTENSIONS
Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021
STATUS
approved