|
%I #6 Jun 04 2022 22:19:33
%S 12,40,60,63,84,112,120,126,132,144,156,204,228,252,276,280,300,315,
%T 325,336,348,351,352,360,372,420,440,444,492,504,516,520,560,564,588,
%U 630,636,650,660,675,680,693,702,708,720,732,760,780,804,819,832,840,852
%N Numbers whose prime indices do not 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 Every sequence can be uniquely split into a sequence of non-overlapping runs. 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 terms together with their prime indices begin:
%e 12: {1,1,2}
%e 40: {1,1,1,3}
%e 60: {1,1,2,3}
%e 63: {2,2,4}
%e 84: {1,1,2,4}
%e 112: {1,1,1,1,4}
%e 120: {1,1,1,2,3}
%e 126: {1,2,2,4}
%e 132: {1,1,2,5}
%e 144: {1,1,1,1,2,2}
%e 156: {1,1,2,6}
%e 204: {1,1,2,7}
%e 228: {1,1,2,8}
%e 252: {1,1,2,2,4}
%e 276: {1,1,2,9}
%e 280: {1,1,1,3,4}
%e 300: {1,1,2,3,3}
%e 315: {2,2,3,4}
%t Select[Range[100],!UnsameQ@@Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]&]
%Y For equal run-sums we have A353833, counted by A304442, nonprime A353834.
%Y The complement is A353838, counted by A353837.
%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 A353852 ranks compositions with all distinct run-sums, counted by A353850.
%Y A353862 gives the greatest run-sum of prime indices, least A353931.
%Y A353866 ranks rucksack partitions, counted by A353864.
%Y Cf. A002110, A071625, A073093, A116608, A118914, A124010, A353861, A353867.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 04 2022
|
