A353931 Least run-sum of the prime indices of n.

0, 1, 2, 2, 3, 1, 4, 3, 4, 1, 5, 2, 6, 1, 2, 4, 7, 1, 8, 2, 2, 1, 9, 2, 6, 1, 6, 2, 10, 1, 11, 5, 2, 1, 3, 2, 12, 1, 2, 3, 13, 1, 14, 2, 3, 1, 15, 2, 8, 1, 2, 2, 16, 1, 3, 3, 2, 1, 17, 2, 18, 1, 4, 6, 3, 1, 19, 2, 2, 1, 20, 3, 21, 1, 2, 2, 4, 1, 22, 3, 8, 1

Offset: 1

Author

Gus Wiseman, Jun 07 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 prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 3.
		

Crossrefs

Positions of first appearances are A008578.

For run-lengths instead of run-sums we have A051904, greatest A051903.

For run-sums and binary expansion we have A144790, greatest A038374.

For run-lengths and binary expansion we have A175597, greatest A043276.

Distinct run-sums are counted by A353835, weak A353861.

The greatest run-sum is given by A353862.

A001222 counts prime factors, distinct A001221.

A005811 counts runs in binary expansion.

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

A124010 gives prime signature, sorted A118914.

A304442 counts partitions with all equal run-sums, compositions A353851.

A353832 represents the operation of taking run-sums of a partition.

A353833 ranks partitions with all equal run sums, nonprime A353834.

A353838 ranks partitions with all distinct run-sums, counted by A353837.

A353840-A353846 pertain to partition run-sum trajectory.

Cf. A067340, A073093, A181819, A300273, A316413, A353839, A353866.

Programs

Mathematica

Table[Min@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]