A383088 Numbers whose multiset of prime indices does not have all equal run-sums.

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Author

Gus Wiseman, Apr 17 2025

Keywords

• nonn

Comments

First differs from A381871 in having 36.

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, sum A056239.

Examples

			The prime indices of 36 are {1,1,2,2}, with run-sums (2,4), so 36 is in the sequence, even though we have the multiset partition {{1,1},{2},{2}} with equal sums.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}
		

Crossrefs

For run-lengths instead of sums we have A059404, distinct A130092.

The complement is A353833, counted by A304442.

For distinct instead of equal run-sums we have A353839.

Partitions of this type are counted by A382076.

Counting and ranking partitions by run-lengths and run-sums:

- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)

- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)

- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)

- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)

- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)

- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.

A353851 counts compositions with a common run-sum, ranks A353848.

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

A382877 counts permutations of prime indices with equal run-sums, zeros A383100.

A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.

Cf. A000720, A006171, A300273, A353861, A353932, A354584, A383014, A383015, A383095, A383097, A383099.

Programs

Mathematica

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