cp's OEIS Frontend

A346698 Sum of the even-indexed parts (even bisection) of the multiset of prime indices of n.

%I A346698 #10 Dec 02 2021 13:00:44
%S A346698 0,0,0,1,0,2,0,1,2,3,0,1,0,4,3,2,0,2,0,1,4,5,0,3,3,6,2,1,0,2,0,2,5,7,
%T A346698 4,3,0,8,6,4,0,2,0,1,2,9,0,2,4,3,7,1,0,4,5,5,8,10,0,4,0,11,2,3,6,2,0,
%U A346698 1,9,3,0,3,0,12,3,1,5,2,0,2,4,13,0,5,7,14,10,6,0,5,6,1,11,15,8,4,0,4,2,4,0,2,0,7,3
%N A346698 Sum of the even-indexed parts (even bisection) of the multiset of prime indices of n.
%C A346698 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.
%H A346698 Antti Karttunen, <a href="/A346698/b346698.txt">Table of n, a(n) for n = 1..20000</a>
%F A346698 a(n) = A056239(n) - A346697(n).
%F A346698 a(n) = A346697(n) - A316524(n).
%F A346698 a(n even omega) = A346699(n).
%F A346698 a(n odd omega) = A346700(n).
%F A346698 A344616(n) = A346699(n) - A346700(n).
%e A346698 The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 = 4.
%e A346698 The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 3 + 4 = 8.
%t A346698 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A346698 Table[Total[Last/@Partition[Append[primeMS[n],0],2]],{n,100}]
%o A346698 (PARI) A346698(n) = if(1==n,0,my(f=factor(n),s=0,p=0); for(k=1,#f~,while(f[k,2], s += (p%2)*primepi(f[k,1]); f[k,2]--; p++)); (s)); \\ _Antti Karttunen_, Nov 30 2021
%Y A346698 Subtracting from the odd version gives A316524 (reverse: A344616).
%Y A346698 The version for standard compositions is A346633 (odd: A209281(n+1)).
%Y A346698 The odd version is A346697.
%Y A346698 The even reverse version is A346699.
%Y A346698 The reverse version is A346700.
%Y A346698 A000120 and A080791 count binary digits 1 and 0, with difference A145037.
%Y A346698 A001414 adds up prime factors, row-sums of A027746.
%Y A346698 A029837 adds up parts of standard compositions (alternating: A124754).
%Y A346698 A056239 adds up prime indices, row-sums of A112798.
%Y A346698 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A346698 A325534 counts separable partitions, ranked by A335433.
%Y A346698 A325535 counts inseparable partitions, ranked by A335448.
%Y A346698 A344606 counts alternating permutations of prime indices.
%Y A346698 Cf. A000070, A000097, A025047, A035363, A120452, A344617, A344653, A344654, A345957, A345958.
%K A346698 nonn
%O A346698 1,6
%A A346698 _Gus Wiseman_, Aug 01 2021
%E A346698 Data section extended up to 105 terms by _Antti Karttunen_, Nov 30 2021