cp's OEIS Frontend

A383091 Numbers whose prime indices have at most one permutation with all equal run-lengths.

%I A383091 #11 Apr 19 2025 19:34:49
%S A383091 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,24,25,27,28,29,31,32,37,
%T A383091 40,41,43,44,45,47,48,49,50,52,53,54,56,59,61,63,64,67,68,71,72,73,75,
%U A383091 76,79,80,81,83,88,89,92,96,97,98,99,101,103,104,107,108,109
%N A383091 Numbers whose prime indices have at most one permutation with all equal run-lengths.
%C A383091 First differs from A359178 (complement A362606) in having 1, 240 and lacking 180.
%C A383091 First differs from A130091 (complement A130092) in having 240 and lacking 360.
%C A383091 First differs from A351294 (complement A351295) in having 240 and lacking 216.
%C A383091 Includes all primes A000040 and prime powers A000961.
%C A383091 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.
%F A383091 Equals A382879 \/ A383112, counted by A382915 + A383094.
%e A383091 The prime indices of 144 are {1,1,1,1,2,2}, with just one permutation with all equal run-lengths (1,1,2,2,1,1), so 144 is in the sequence.
%e A383091 The prime indices of 240 are {1,1,1,1,2,3}, which have no permutation with all equal run-lengths, so 240 is in the sequence.
%e A383091 The terms together with their prime indices begin:
%e A383091    1: {}
%e A383091    2: {1}
%e A383091    3: {2}
%e A383091    4: {1,1}
%e A383091    5: {3}
%e A383091    7: {4}
%e A383091    8: {1,1,1}
%e A383091    9: {2,2}
%e A383091   11: {5}
%e A383091   12: {1,1,2}
%e A383091   13: {6}
%e A383091   16: {1,1,1,1}
%e A383091   17: {7}
%e A383091   18: {1,2,2}
%e A383091   19: {8}
%e A383091   20: {1,1,3}
%e A383091   23: {9}
%e A383091   24: {1,1,1,2}
%t A383091 Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]]<=1&]
%Y A383091 These are positions of zeros and ones in A382857, just zeros A382879, just ones A383112.
%Y A383091 The complement for run-sums instead of lengths is A383015, counted by A383097.
%Y A383091 The complement is A383089, counted by A383090.
%Y A383091 Partitions of this type are counted by A383092, just zero A382915, just one A383094.
%Y A383091 For run-sums instead of lengths we have A383099 \/ A383100, counted by A383095 + A383096.
%Y A383091 A047966 counts partitions with equal run-lengths, compositions A329738.
%Y A383091 A056239 adds up prime indices, row sums of A112798.
%Y A383091 A098859 counts partitions with distinct run-lengths, ranks A130091.
%Y A383091 A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
%Y A383091 Cf. A000720, A000961, A001221, A001222, A048767, A351294, A353744, A353833, A381435, A382771, A382877, A383113.
%K A383091 nonn
%O A383091 1,2
%A A383091 _Gus Wiseman_, Apr 18 2025