cp's OEIS Frontend

A357630 Skew-alternating sum of the prime indices of n.

%I A357630 #6 Oct 09 2022 09:41:55
%S A357630 0,1,2,0,3,-1,4,-1,0,-2,5,-2,6,-3,-1,0,7,-3,8,-3,-2,-4,9,1,0,-5,-2,-4,
%T A357630 10,-4,11,1,-3,-6,-1,0,12,-7,-4,2,13,-5,14,-5,-3,-8,15,2,0,-5,-5,-6,
%U A357630 16,-1,-2,3,-6,-9,17,1,18,-10,-4,0,-3,-6,19,-7,-7,-6,20
%N A357630 Skew-alternating sum of the prime indices of n.
%C A357630 We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
%C A357630 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.
%e A357630 The prime indices of 525 are {2,3,3,4} so a(525) = 2 - 3 - 3 + 4 = 0.
%t A357630 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357630 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357630 Table[skats[primeMS[n]],{n,30}]
%Y A357630 The original alternating sum is A316524, reverse A344616.
%Y A357630 The reverse version is A357634.
%Y A357630 The half-alternating form is A357629, reverse A357633.
%Y A357630 Positions of zeros are A357632, reverse A357636.
%Y A357630 The version for standard compositions is A357623, reverse A357624.
%Y A357630 These partitions are counted by A357638, half A357637.
%Y A357630 A056239 adds up prime indices, row sums of A112798.
%Y A357630 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357630 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357630 A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
%Y A357630 Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621, A357622, A357625, A357626, A357635, A357640.
%K A357630 sign
%O A357630 1,3
%A A357630 _Gus Wiseman_, Oct 09 2022