A353839 Numbers whose prime indices do not have all distinct run-sums.

12, 40, 60, 63, 84, 112, 120, 126, 132, 144, 156, 204, 228, 252, 276, 280, 300, 315, 325, 336, 348, 351, 352, 360, 372, 420, 440, 444, 492, 504, 516, 520, 560, 564, 588, 630, 636, 650, 660, 675, 680, 693, 702, 708, 720, 732, 760, 780, 804, 819, 832, 840, 852

Offset: 1

Author

Gus Wiseman, Jun 04 2022

Keywords

• nonn

Comments

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.

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).

Examples

			The terms together with their prime indices begin:
   12: {1,1,2}
   40: {1,1,1,3}
   60: {1,1,2,3}
   63: {2,2,4}
   84: {1,1,2,4}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  144: {1,1,1,1,2,2}
  156: {1,1,2,6}
  204: {1,1,2,7}
  228: {1,1,2,8}
  252: {1,1,2,2,4}
  276: {1,1,2,9}
  280: {1,1,1,3,4}
  300: {1,1,2,3,3}
  315: {2,2,3,4}
		

Crossrefs

For equal run-sums we have A353833, counted by A304442, nonprime A353834.

The complement is A353838, counted by A353837.

A001222 counts prime factors, distinct A001221.

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

A098859 counts partitions with distinct multiplicities, ranked by A130091.

A165413 counts distinct run-sums in binary expansion.

A300273 ranks collapsible partitions, counted by A275870.

A351014 counts distinct runs in standard compositions.

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

A353840-A353846 pertain to partition run-sum trajectory.

A353852 ranks compositions with all distinct run-sums, counted by A353850.

A353862 gives the greatest run-sum of prime indices, least A353931.

A353866 ranks rucksack partitions, counted by A353864.

Cf. A002110, A071625, A073093, A116608, A118914, A124010, A353861, A353867.

Programs

Mathematica

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