A353838 Numbers whose prime indices have all distinct run-sums.

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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Author

Gus Wiseman, May 23 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.

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).

Examples

			The prime indices of 180 are {1,1,2,2,3}, with run-sums (2,4,3), so 180 is in the sequence.
The prime indices of 315 are {2,2,3,4}, with run-sums (4,3,4), so 315 is not in the sequence.
		

Crossrefs

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

These partitions are counted by A353837.

The complement is A353839.

The version for compositions is A353852, counted by A353850.

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

The weak case is A353866, counted by A353864.

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.

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

Programs

Mathematica

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