|
%I #6 Jan 17 2020 17:41:41
%S 0,1,1,2,2,2,3,3,2,3,6,3,7,4,3,4,10,3,11,4,4,7,14,4,4,8,3,5,19,4,20,5,
%T 7,11,5,4,25,12,8,5,28,5,29,8,4,15,32,5,6,5,11,9,37,4,8,6,12,20,42,5,
%U 43,21,5,6,9,8,48,12,15,6,51,5,52,26,5,13,9,9
%N Sum of prime factors minus sum of prime indices of n.
%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 Totally additive with a(prime(k)) = prime(k) - k = A014689(k).
%F a(n) = A001414(n) - A056239(n).
%e The prime factors of 12 are {2,2,3}, while the prime indices are {1,1,2}, so a(12) = 7 - 4 = 3.
%t Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k*(p-PrimePi[p])]],{n,30}]
%Y The number of k's is A331387(k) = sum of k-th column of A331385.
%Y The sum of prime factors of n is A001414(n).
%Y The sum of prime indices of n is A056239(n).
%Y Numbers divisible by the sum of their prime factors are A036844.
%Y Sum of prime factors is divisible by sum of prime indices: A331380
%Y Product of prime indices equals sum of prime factors: A331384.
%Y Cf. A000040, A000720, A001222, A014689, A056239, A112798, A301987, A318995, A325036, A330953, A331378, A331379, A331416, A331418.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jan 17 2020
|
