A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

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

Offset: 1

Author

Gus Wiseman, Jul 06 2024

Keywords

• nonn

Comments

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

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.

Examples

			The prime indices of 24 are {1,1,1,2}, with permutations such as (1,1,2,1) whose run-compression sums to 4, so a(24) = 4.
The prime indices of 216 are {1,1,1,2,2,2}, with permutations such as (1,2,1,2,1,2) whose run-compression sums to 9, so a(216) = 9.
		

Crossrefs

Positions of first appearances are 1 followed by the primes A000040.

Positions of 1 are A000079 (powers of two) except 1.

Positions of 2 are A000244 (powers of three) except 1.

Positions of 3 are {6} U A000351 (six or powers of five) except 1.

For number of runs instead of sum of run-compression we have A373957.

For prime factors instead of indices we have A374250.

A001221 counts distinct prime factors, A001222 with multiplicity.

A003242 counts run-compressed compositions, i.e., anti-runs.

A007947 (squarefree kernel) represents run-compression of multisets.

A008480 counts permutations of prime factors (or prime indices).

A056239 adds up prime indices, row sums of A112798.

A116861 counts partitions by sum of run-compression.

A304038 lists run-compression of prime indices, sum A066328.

A335433 lists numbers whose prime indices are separable, complement A335448.

A373949 counts compositions by sum of run-compression, opposite A373951.

A374251 run-compresses standard compositions, sum A373953, rank A373948.

Cf. A001223, A001414, A037201, A116608, A124767, A316413, A333755, A340387, A344415, A373954, A374246, A374247.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

Formula

a(n) = A056239(n) iff n belongs to A335433 (the separable case), complement A335448.