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A357851
Numbers k such that the half-alternating sum of the prime indices of k is 1.
1
2, 8, 18, 32, 45, 50, 72, 98, 105, 128, 162, 180, 200, 231, 242, 275, 288, 338, 392, 420, 429, 450, 455, 512, 578, 648, 663, 720, 722, 800, 833, 882, 924, 935, 968, 969, 1050, 1058, 1100, 1125, 1152, 1235, 1250, 1311, 1352, 1458, 1463, 1568, 1680, 1682, 1716
OFFSET
1,1
COMMENTS
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 - ...
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.
EXAMPLE
The terms together with their prime indices begin:
2: {1}
8: {1,1,1}
18: {1,2,2}
32: {1,1,1,1,1}
45: {2,2,3}
50: {1,3,3}
72: {1,1,1,2,2}
98: {1,4,4}
105: {2,3,4}
128: {1,1,1,1,1,1,1}
162: {1,2,2,2,2}
180: {1,1,2,2,3}
200: {1,1,1,3,3}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Select[Range[1000], halfats[primeMS[#]]==1&]
CROSSREFS
The version for k = 0 is A357631, standard compositions A357625-A357626.
The version for original alternating sum is A001105.
Positions of ones in A357629, reverse A357633.
The skew version for k = 0 is A357632, reverse A357636.
Partitions with these Heinz numbers are counted by A035444, skew A035544.
The reverse version is A357635, k = 0 version A000583.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even-length A357642.
Sequence in context: A293296 A055044 A356209 * A067051 A074629 A209303
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 28 2022
STATUS
approved