cp's OEIS Frontend

A374255 Sum of prime factors of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime factors of n.

%I A374255 #8 Apr 25 2025 23:40:18
%S A374255 0,0,0,2,0,0,0,4,3,0,0,0,0,0,0,6,0,0,0,0,0,0,0,2,5,0,6,0,0,0,0,8,0,0,
%T A374255 0,0,0,0,0,2,0,0,0,0,0,0,0,4,7,0,0,0,0,3,0,2,0,0,0,0,0,0,0,10,0,0,0,0,
%U A374255 0,0,0,0,0,0,0,0,0,0,0,4,9,0,0,0,0,0,0
%N A374255 Sum of prime factors of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime factors of n.
%C A374255 Contains no ones.
%C A374255 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 A374255 a(n) = A001414(n) - A374250(n).
%e A374255 The prime factors of 96 are {2,2,2,2,2,3}, with sum 13, and we have permutations such as (2,2,2,2,3,2), with run-compression (2,3,2), with sum 7, so a(96) = 13 - 7 = 6.
%t A374255 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A374255 Table[Total[prifacs[n]]-Max@@(Total[First/@Split[#]]& /@ Permutations[prifacs[n]]),{n,100}]
%Y A374255 Positions of first appearances are A280286.
%Y A374255 For least instead of greatest sum of run-compression we have A280292.
%Y A374255 Positions of zeros are A335433 (separable).
%Y A374255 Positions of positive terms are A335448 (inseparable).
%Y A374255 For prime indices instead of factors we have A374248.
%Y A374255 This is an opposite version of A374250, for prime indices A373956.
%Y A374255 A001221 counts distinct prime factors, A001222 with multiplicity.
%Y A374255 A003242 counts run-compressed compositions, i.e., anti-runs.
%Y A374255 A007947 (squarefree kernel) represents run-compression of multisets.
%Y A374255 A008480 counts permutations of prime factors.
%Y A374255 A027746 lists prime factors, row-sums A001414.
%Y A374255 A027748 is run-compression of prime factors, row-sums A008472.
%Y A374255 A056239 adds up prime indices, row sums of A112798.
%Y A374255 A116861 counts partitions by sum of run-compression.
%Y A374255 A304038 is run-compression of prime indices, row-sums A066328.
%Y A374255 A373949 counts compositions by sum of run-compression, opposite A373951.
%Y A374255 A373957 gives greatest number of runs in a permutation of prime factors.
%Y A374255 A374251 run-compresses standard compositions, sum A373953, rank A373948.
%Y A374255 A374252 counts permutations of prime factors by number of runs.
%Y A374255 Cf. A000040, A026549, A037201, A046660, A124767, A246655, A333755, A373954, A374246, A374247.
%K A374255 nonn
%O A374255 1,4
%A A374255 _Gus Wiseman_, Jul 10 2024