cp's OEIS Frontend

A357851 Numbers k such that the half-alternating sum of the prime indices of k is 1.

%I A357851 #5 Oct 29 2022 09:10:22
%S A357851 2,8,18,32,45,50,72,98,105,128,162,180,200,231,242,275,288,338,392,
%T A357851 420,429,450,455,512,578,648,663,720,722,800,833,882,924,935,968,969,
%U A357851 1050,1058,1100,1125,1152,1235,1250,1311,1352,1458,1463,1568,1680,1682,1716
%N A357851 Numbers k such that the half-alternating sum of the prime indices of k is 1.
%C A357851 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 A357851 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 A357851 The terms together with their prime indices begin:
%e A357851      2: {1}
%e A357851      8: {1,1,1}
%e A357851     18: {1,2,2}
%e A357851     32: {1,1,1,1,1}
%e A357851     45: {2,2,3}
%e A357851     50: {1,3,3}
%e A357851     72: {1,1,1,2,2}
%e A357851     98: {1,4,4}
%e A357851    105: {2,3,4}
%e A357851    128: {1,1,1,1,1,1,1}
%e A357851    162: {1,2,2,2,2}
%e A357851    180: {1,1,2,2,3}
%e A357851    200: {1,1,1,3,3}
%t A357851 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357851 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357851 Select[Range[1000],halfats[primeMS[#]]==1&]
%Y A357851 The version for k = 0 is A357631, standard compositions A357625-A357626.
%Y A357851 The version for original alternating sum is A001105.
%Y A357851 Positions of ones in A357629, reverse A357633.
%Y A357851 The skew version for k = 0 is A357632, reverse A357636.
%Y A357851 Partitions with these Heinz numbers are counted by A035444, skew A035544.
%Y A357851 The reverse version is A357635, k = 0 version A000583.
%Y A357851 A056239 adds up prime indices, row sums of A112798.
%Y A357851 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A357851 A351005 = alternately equal and unequal partitions, compositions A357643.
%Y A357851 A351006 = alternately unequal and equal partitions, compositions A357644.
%Y A357851 A357641 counts comps w/ half-alt sum 0, even-length A357642.
%Y A357851 Cf. A003963, A053251, A055932, A345958, A357621-A357624, A357630, A357634, A357637, A357639, A357640.
%K A357851 nonn
%O A357851 1,1
%A A357851 _Gus Wiseman_, Oct 28 2022