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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.

%I A374248 #10 Apr 25 2025 23:40:15
%S A374248 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 A374248 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 A374248 0,0,0,0,0,0,0,0,0,0,0,2,6,0,0,0,0,0,0
%N 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.
%C A374248 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 A374248 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 A374248 a(n) = A056239(n) - A373956(n).
%e A374248 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 A374248 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A374248 Table[Total[prix[n]]-Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]
%Y A374248 Positions of zeros are A335433 (separable).
%Y A374248 Positions of positive terms are A335448 (inseparable).
%Y A374248 This is an opposite version of A373956, for prime factors A374250.
%Y A374248 For prime factors instead of indices we have A374255.
%Y A374248 A001221 counts distinct prime factors, A001222 with multiplicity.
%Y A374248 A003242 counts run-compressed compositions, i.e., anti-runs.
%Y A374248 A007947 (squarefree kernel) represents run-compression of multisets.
%Y A374248 A008480 counts permutations of prime factors.
%Y A374248 A027746 lists prime factors, row-sums A001414.
%Y A374248 A027748 is run-compression of prime factors, row-sums A008472.
%Y A374248 A056239 adds up prime indices, row-sums of A112798.
%Y A374248 A116861 counts partitions by sum of run-compression.
%Y A374248 A304038 is run-compression of prime indices, row-sums A066328.
%Y A374248 A373949 counts compositions by sum of run-compression, opposite A373951.
%Y A374248 A373957 gives greatest number of runs in a permutation of prime factors.
%Y A374248 A374251 run-compresses standard compositions, sum A373953, rank A373948.
%Y A374248 A374252 counts permutations of prime factors by number of runs.
%Y A374248 Cf. A000040, A026549, A037201, A046660, A124767, A280286, A280292, A333755, A373954, A374246, A374247.
%K A374248 nonn
%O A374248 1,8
%A A374248 _Gus Wiseman_, Jul 10 2024