cp's OEIS Frontend

A374250 Greatest sum of run-compression of a permutation of the prime factors of n.

%I A374250 #9 Jul 13 2024 07:43:35
%S A374250 0,2,3,2,5,5,7,2,3,7,11,7,13,9,8,2,17,8,19,9,10,13,23,7,5,15,3,11,29,
%T A374250 10,31,2,14,19,12,10,37,21,16,9,41,12,43,15,11,25,47,7,7,12,20,17,53,
%U A374250 8,16,11,22,31,59,12,61,33,13,2,18,16,67,21,26,14,71
%N A374250 Greatest sum of run-compression of a permutation of the prime factors of n.
%C A374250 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).
%F A374250 a(n) = A001414(n) iff n belongs to A335433 (the separable case, complement A335448), row-sums of A027746.
%e A374250 The prime factors of 24 are {2,2,2,3}, with permutations such as (2,2,3,2) whose run-compression sums to 7, so a(24) = 7.
%e A374250 The prime factors of 216 are {2,2,2,3,3,3}, with permutations such as (2,3,2,3,2,3) whose run-compression sums to 15, so a(216) = 15.
%t A374250 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A374250 Table[Max@@(Total[First/@Split[#]]& /@ Permutations[prifacs[n]]),{n,100}]
%Y A374250 Positions of 2 are A000079 (powers of two) except 1.
%Y A374250 Positions of 3 are A000244 (powers of three) except 1.
%Y A374250 For least instead of greatest sum of run-compression we have A008472.
%Y A374250 For prime indices instead of factors we have A373956.
%Y A374250 For number of runs instead of sum of run-compression we have A373957.
%Y A374250 A001221 counts distinct prime factors, A001222 with multiplicity.
%Y A374250 A003242 counts run-compressed compositions, i.e., anti-runs.
%Y A374250 A007947 (squarefree kernel) represents run-compression of multisets.
%Y A374250 A008480 counts permutations of prime factors (or prime indices).
%Y A374250 A056239 adds up prime indices, row sums of A112798.
%Y A374250 A116861 counts partitions by sum of run-compression.
%Y A374250 A304038 lists run-compression of prime indices, sum A066328.
%Y A374250 A335433 lists numbers whose prime indices are separable, complement A335448.
%Y A374250 A373949 counts compositions by sum of run-compression, opposite A373951.
%Y A374250 A374251 run-compresses standard compositions, sum A373953, rank A373948.
%Y A374250 Cf. A000040, A001223, A001414, A037201, A116608, A124767, A333755, A373954, A374246, A374247, A374252.
%K A374250 nonn
%O A374250 1,2
%A A374250 _Gus Wiseman_, Jul 09 2024