A353839 - OEIS

%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