A353838 - OEIS

%I #5 May 24 2022 22:33:59

%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,

%T 28,29,30,31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,

%U 52,53,54,55,56,57,58,59,61,62,64,65,66,67,68,69,70,71

%N Numbers whose prime indices have all distinct run-sums.

%C 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.

%C The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

%e The prime indices of 180 are {1,1,2,2,3}, with run-sums (2,4,3), so 180 is in the sequence.

%e The prime indices of 315 are {2,2,3,4}, with run-sums (4,3,4), so 315 is not in the sequence.

%t Select[Range[100],UnsameQ@@Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]&]

%Y The version for all equal run-sums is A353833, counted by A304442.

%Y These partitions are counted by A353837.

%Y The complement is A353839.

%Y The version for compositions is A353852, counted by A353850.

%Y The greatest run-sum is given by A353862, least A353931.

%Y The weak case is A353866, counted by A353864.

%Y A001222 counts prime factors, distinct A001221.

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A098859 counts partitions with distinct multiplicities, ranked by A130091.

%Y A165413 counts distinct run-sums in binary expansion.

%Y A300273 ranks collapsible partitions, counted by A275870.

%Y A351014 counts distinct runs in standard compositions.

%Y A353832 represents taking run-sums of a partition, compositions A353847.

%Y A353840-A353846 pertain to partition run-sum trajectory.

%Y Cf. A002110, A071625, A073093, A118914, A124010, A353834, A353861, A353867.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 23 2022