A374248 Sum of prime indices of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime indices of n.

0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Jul 10 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 96 are {1,1,1,1,1,2}, with sum 7, and we have permutations such as (1,1,1,1,2,1), with run-compression (1,2,1), with sum 4, so a(96) = 7 - 4 = 3.
		

Crossrefs

Positions of zeros are A335433 (separable).

Positions of positive terms are A335448 (inseparable).

This is an opposite version of A373956, for prime factors A374250.

For prime factors instead of indices we have A374255.

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.

A027746 lists prime factors, row-sums A001414.

A027748 is run-compression of prime factors, row-sums A008472.

A056239 adds up prime indices, row-sums of A112798.

A116861 counts partitions by sum of run-compression.

A304038 is run-compression of prime indices, row-sums A066328.

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

A373957 gives greatest number of runs in a permutation of prime factors.

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

A374252 counts permutations of prime factors by number of runs.

Cf. A000040, A026549, A037201, A046660, A124767, A280286, A280292, A333755, A373954, A374246, A374247.

Programs

Mathematica

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

Formula

a(n) = A056239(n) - A373956(n).