A374248 - OEIS

%I #7 Sep 16 2024 08:45:46

%S 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,

%T 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,

%U 0,0,0,0,0,0,0,0,0,0,0,2,6,0,0,0,0,0,0

%N 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.

%C 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).

%C 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.

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

%e 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.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Total[prix[n]]-Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

%Y Positions of zeros are A335433 (separable).

%Y Positions of positive terms are A335448 (inseparable).

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

%Y For prime factors instead of indices we have A374255.

%Y A001221 counts distinct prime factors, A001222 with multiplicity.

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

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

%Y A008480 counts permutations of prime factors.

%Y A027746 lists prime factors, row-sums A001414.

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

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

%Y A116861 counts partitions by sum of run-compression.

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

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

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

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

%Y A374252 counts permutations of prime factors by number of runs.


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

%K nonn

%O 1,8

%A _Gus Wiseman_, Jul 10 2024