A357629 - OEIS

%I #6 Oct 09 2022 09:42:00

%S 0,1,2,2,3,3,4,1,4,4,5,0,6,5,5,0,7,1,8,-1,6,6,9,-1,6,7,2,-2,10,0,11,1,

%T 7,8,7,-2,12,9,8,-2,13,-1,14,-3,1,10,15,2,8,1,9,-4,16,-1,8,-3,10,11,

%U 17,-3,18,12,0,2,9,-2,19,-5,11,0,20,1,21,13,2,-6

%N Half-alternating sum of the prime indices of n.

%C We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

%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.

%e The prime indices of 525 are {2,3,3,4} so a(525) = 2 + 3 - 3 - 4 = -2.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];

%t Table[halfats[primeMS[n]],{n,30}]

%Y The original alternating sum is A316524, reverse A344616.

%Y The version for standard compositions is A357621, reverse A357622.

%Y The skew-alternating form is A357630, reverse A357634.

%Y Positions of zeros are A357631, reverse A357635.

%Y The reverse version is A357633.

%Y These partitions are counted by A357637, skew A357638.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A351005 = alternately equal and unequal partitions, compositions A357643.

%Y A351006 = alternately unequal and equal partitions, compositions A357644.

%Y A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.

%Y Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357623-A357626, A357632, A357636, A357640.

%K sign

%O 1,3

%A _Gus Wiseman_, Oct 08 2022