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A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

%I A383088 #9 Apr 18 2025 09:57:41
%S A383088 6,10,14,15,18,20,21,22,24,26,28,30,33,34,35,36,38,39,42,44,45,46,48,
%T A383088 50,51,52,54,55,56,57,58,60,62,65,66,68,69,70,72,74,75,76,77,78,80,82,
%U A383088 84,85,86,87,88,90,91,92,93,94,95,96,98,99,100,102,104,105
%N A383088 Numbers whose multiset of prime indices does not have all equal run-sums.
%C A383088 First differs from A381871 in having 36.
%C A383088 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 A383088 The prime indices of 36 are {1,1,2,2}, with run-sums (2,4), so 36 is in the sequence, even though we have the multiset partition {{1,1},{2},{2}} with equal sums.
%e A383088 The terms together with their prime indices begin:
%e A383088     6: {1,2}
%e A383088    10: {1,3}
%e A383088    14: {1,4}
%e A383088    15: {2,3}
%e A383088    18: {1,2,2}
%e A383088    20: {1,1,3}
%e A383088    21: {2,4}
%e A383088    22: {1,5}
%e A383088    24: {1,1,1,2}
%e A383088    26: {1,6}
%e A383088    28: {1,1,4}
%e A383088    30: {1,2,3}
%e A383088    33: {2,5}
%e A383088    34: {1,7}
%e A383088    35: {3,4}
%e A383088    36: {1,1,2,2}
%e A383088    38: {1,8}
%e A383088    39: {2,6}
%e A383088    42: {1,2,4}
%e A383088    44: {1,1,5}
%e A383088    45: {2,2,3}
%e A383088    46: {1,9}
%t A383088 Select[Range[100], !SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]
%Y A383088 For run-lengths instead of sums we have A059404, distinct A130092.
%Y A383088 The complement is A353833, counted by A304442.
%Y A383088 For distinct instead of equal run-sums we have A353839.
%Y A383088 Partitions of this type are counted by A382076.
%Y A383088 Counting and ranking partitions by run-lengths and run-sums:
%Y A383088 - constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
%Y A383088 - distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
%Y A383088 - weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
%Y A383088 - weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
%Y A383088 - strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
%Y A383088 - strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
%Y A383088 A001222 counts prime factors, distinct A001221.
%Y A383088 A056239 adds up prime indices, row sums of A112798.
%Y A383088 A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
%Y A383088 A353851 counts compositions with a common run-sum, ranks A353848.
%Y A383088 A353862 gives the greatest run-sum of prime indices, least A353931.
%Y A383088 A382877 counts permutations of prime indices with equal run-sums, zeros A383100.
%Y A383088 A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
%Y A383088 Cf. A000720, A006171, A300273, A353861, A353932, A354584, A383014, A383015, A383095, A383097, A383099.
%K A383088 nonn
%O A383088 1,1
%A A383088 _Gus Wiseman_, Apr 17 2025