A357632 - OEIS

%I #9 Oct 10 2022 20:47:04

%S 1,4,9,16,25,36,49,64,81,90,100,121,144,169,196,210,225,256,289,324,

%T 360,361,400,441,462,484,525,529,550,576,625,676,729,784,840,841,858,

%U 900,910,961,1024,1089,1155,1156,1225,1296,1326,1369,1440,1444,1521,1600

%N Numbers k such that the skew-alternating sum of the prime indices of k is 0.

%C 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 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 terms together with their prime indices begin:

%e      1: {}

%e      4: {1,1}

%e      9: {2,2}

%e     16: {1,1,1,1}

%e     25: {3,3}

%e     36: {1,1,2,2}

%e     49: {4,4}

%e     64: {1,1,1,1,1,1}

%e     81: {2,2,2,2}

%e     90: {1,2,2,3}

%e    100: {1,1,3,3}

%e    121: {5,5}

%e    144: {1,1,1,1,2,2}

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

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

%t Select[Range[1000],skats[primeMS[#]]==0&]

%Y The version for original alternating sum is A000290.

%Y The version for standard compositions is A357627, reverse A357628.

%Y Positions of zeros in A357630, reverse A357634.

%Y The half-alternating form is A357631, reverse A357635.

%Y The reverse version is A357636.

%Y These partitions are counted by A357640, half A357639.

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

%Y A316524 gives alternating sum of prime indices, reverse A344616.

%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, even A357642.

%Y Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 09 2022