cp's OEIS Frontend

A357632 Numbers k such that the skew-alternating sum of the prime indices of k is 0.

%I A357632 #9 Oct 10 2022 20:47:04
%S A357632 1,4,9,16,25,36,49,64,81,90,100,121,144,169,196,210,225,256,289,324,
%T A357632 360,361,400,441,462,484,525,529,550,576,625,676,729,784,840,841,858,
%U A357632 900,910,961,1024,1089,1155,1156,1225,1296,1326,1369,1440,1444,1521,1600
%N A357632 Numbers k such that the skew-alternating sum of the prime indices of k is 0.
%C A357632 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 A357632 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 A357632 The terms together with their prime indices begin:
%e A357632      1: {}
%e A357632      4: {1,1}
%e A357632      9: {2,2}
%e A357632     16: {1,1,1,1}
%e A357632     25: {3,3}
%e A357632     36: {1,1,2,2}
%e A357632     49: {4,4}
%e A357632     64: {1,1,1,1,1,1}
%e A357632     81: {2,2,2,2}
%e A357632     90: {1,2,2,3}
%e A357632    100: {1,1,3,3}
%e A357632    121: {5,5}
%e A357632    144: {1,1,1,1,2,2}
%t A357632 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357632 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357632 Select[Range[1000],skats[primeMS[#]]==0&]
%Y A357632 The version for original alternating sum is A000290.
%Y A357632 The version for standard compositions is A357627, reverse A357628.
%Y A357632 Positions of zeros in A357630, reverse A357634.
%Y A357632 The half-alternating form is A357631, reverse A357635.
%Y A357632 The reverse version is A357636.
%Y A357632 These partitions are counted by A357640, half A357639.
%Y A357632 A056239 adds up prime indices, row sums of A112798.
%Y A357632 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A357632 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357632 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357632 A357641 counts comps w/ half-alt sum 0, even A357642.
%Y A357632 Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.
%K A357632 nonn
%O A357632 1,2
%A A357632 _Gus Wiseman_, Oct 09 2022