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A335452
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Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.
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58
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1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 0, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2
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OFFSET
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1,6
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COMMENTS
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The first term that is not a factorial number is a(180) = 12.
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.
A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Feb 03 2021
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LINKS
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EXAMPLE
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The a(n) separations for n = 2, 6, 30, 180:
(1) (12) (123) (12123)
(21) (132) (12132)
(213) (12312)
(231) (12321)
(312) (13212)
(321) (21213)
(21231)
(21312)
(21321)
(23121)
(31212)
(32121)
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MATHEMATICA
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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, 100}]
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PROG
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(PARI)
F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i, t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
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CROSSREFS
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Permutations of prime indices are counted by A008480.
Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, A335489.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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