cp's OEIS Frontend

A383110 Numbers whose prime indices have a permutation with all equal run-sums.

%I A383110 #6 Apr 22 2025 09:10:03
%S A383110 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,25,27,29,31,32,36,37,40,41,43,
%T A383110 47,48,49,53,59,61,63,64,67,71,73,79,81,83,89,97,101,103,107,109,112,
%U A383110 113,121,125,127,128,131,137,139,144,149,151,157,163,167,169,173
%N A383110 Numbers whose prime indices have a permutation with all equal run-sums.
%C A383110 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 A383110 Equals A383015 \/ A383099, counted by A353851 \/ A383095.
%e A383110 The prime indices of 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), so 144 is in the sequence.
%e A383110 The terms together with their prime indices begin:
%e A383110    1: {}
%e A383110    2: {1}
%e A383110    3: {2}
%e A383110    4: {1,1}
%e A383110    5: {3}
%e A383110    7: {4}
%e A383110    8: {1,1,1}
%e A383110    9: {2,2}
%e A383110   11: {5}
%e A383110   12: {1,1,2}
%e A383110   13: {6}
%e A383110   16: {1,1,1,1}
%e A383110   17: {7}
%e A383110   19: {8}
%e A383110   23: {9}
%e A383110   25: {3,3}
%e A383110   27: {2,2,2}
%e A383110   29: {10}
%e A383110   31: {11}
%e A383110   32: {1,1,1,1,1}
%e A383110   36: {1,1,2,2}
%e A383110   37: {12}
%t A383110 Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]>0&]
%Y A383110 For distinct run-sums we appear to have complement of A381636 (counted by A381717).
%Y A383110 These are the positions of positive terms in A382877.
%Y A383110 For run-lengths instead of sums we have complement of A382879, counted by A383013.
%Y A383110 For more than one choice we have A383015.
%Y A383110 Partitions of this type are counted by A383098.
%Y A383110 For a unique choice we have A383099, counted by A383095.
%Y A383110 The complement is A383100, counted by A383096.
%Y A383110 A056239 adds up prime indices, row sums of A112798.
%Y A383110 A304442 counts partitions with equal run-sums, ranks A353833.
%Y A383110 A353851 counts compositions with equal run-sums, ranks A353848.
%Y A383110 Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
%Y A383110 Cf. A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383097.
%K A383110 nonn
%O A383110 1,2
%A A383110 _Gus Wiseman_, Apr 20 2025