cp's OEIS Frontend

A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.

%I A382857 #10 Apr 21 2025 10:47:15
%S A382857 1,1,1,1,1,1,2,1,1,1,2,1,1,1,2,2,1,1,1,1,1,2,2,1,0,1,2,1,1,1,6,1,1,2,
%T A382857 2,2,4,1,2,2,0,1,6,1,1,1,2,1,0,1,1,2,1,1,0,2,0,2,2,1,6,1,2,1,1,2,6,1,
%U A382857 1,2,6,1,1,1,2,1,1,2,6,1,0,1,2,1,6,2,2
%N A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.
%C A382857 The first x with a(x) > 1 but A382771(x) > 0 is a(216) = 4, A382771(216) = 4.
%C A382857 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, sum A056239.
%e A382857 The prime indices of 216 are {1,1,1,2,2,2} and we have permutations:
%e A382857   (1,1,1,2,2,2)
%e A382857   (1,2,1,2,1,2)
%e A382857   (2,1,2,1,2,1)
%e A382857   (2,2,2,1,1,1)
%e A382857 so a(216) = 4.
%e A382857 The prime indices of 25920 are {1,1,1,1,1,1,2,2,2,2,3} and we have permutations:
%e A382857   (1,2,1,2,1,2,1,2,1,3,1)
%e A382857   (1,2,1,2,1,2,1,3,1,2,1)
%e A382857   (1,2,1,2,1,3,1,2,1,2,1)
%e A382857   (1,2,1,3,1,2,1,2,1,2,1)
%e A382857   (1,3,1,2,1,2,1,2,1,2,1)
%e A382857 so a(25920) = 5.
%t A382857 Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]], SameQ@@Length/@Split[#]&]],{n,0,100}]
%Y A382857 The restriction to signature representatives (A181821) is A382858, distinct A382773.
%Y A382857 The restriction to factorials is A335407, distinct A382774.
%Y A382857 For distinct instead of equal run-lengths we have A382771.
%Y A382857 For run-sums instead of run-lengths we have A382877, distinct A382876.
%Y A382857 Positions of first appearances are A382878.
%Y A382857 Positions of 0 are A382879.
%Y A382857 Positions of terms > 1 are A383089.
%Y A382857 Positions of 1 are A383112.
%Y A382857 A003963 gives product of prime indices.
%Y A382857 A005811 counts runs in binary expansion.
%Y A382857 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A382857 A056239 adds up prime indices, row sums of A112798.
%Y A382857 A239455 counts Look-and-Say partitions, ranks A351294.
%Y A382857 A304442 counts partitions with equal run-sums, ranks A353833.
%Y A382857 A164707 lists numbers whose binary expansion has all equal run-lengths, distinct A328592.
%Y A382857 A353744 ranks compositions with equal run-lengths, counted by A329738.
%Y A382857 Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A238130, A238279, A351201, A351293, A351295.
%K A382857 nonn
%O A382857 0,7
%A A382857 _Gus Wiseman_, Apr 09 2025