A353833 Numbers whose multiset of prime indices has all equal run-sums.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179

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 12 are {1,1,2}, with run-sums (2,2), so 12 is in the sequence.
		

Links

• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Crossrefs

For parts instead of run-sums we have A000961, counted by A000005.

For run-lengths instead of run-sums we have A072774, counted by A047966.

These partitions are counted by A304442.

These are the positions of powers of primes in A353832.

The restriction to nonprimes is A353834.

For distinct instead of equal run-sums we have A353838, counted by A353837.

The version for compositions is A353848, counted by A353851.

A001222 counts prime factors, distinct A001221.

A005811 counts runs in binary expansion, distinct run-lengths A165413.

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

A124010 gives prime signature, sorted A118914.

A300273 ranks collapsible partitions, counted by A275870.

A353835 counts distinct run-sums of prime indices, weak A353861.

A353840-A353846 deal with iterated run-sums for partitions.

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

A353866 ranks rucksack partitions, counted by A353864.

Cf. A007947, A071625, A073093, A181819, A238279, A304660, A323014, A333755, A353839.

Programs

Mathematica

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