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A345962
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Numbers whose prime indices have alternating sum -2.
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5
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10, 21, 40, 55, 84, 90, 91, 160, 187, 189, 210, 220, 247, 250, 336, 360, 364, 391, 462, 490, 495, 525, 551, 640, 713, 748, 756, 810, 819, 840, 858, 880, 988, 1000, 1029, 1073, 1155, 1210, 1271, 1326, 1344, 1375, 1440, 1456, 1564, 1591, 1683, 1690, 1701, 1848
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OFFSET
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1,1
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COMMENTS
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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.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also numbers with even Omega (A001222) and exactly two odd conjugate prime indices. The case of odd Omega is A345960, and the union is A345961.
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LINKS
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EXAMPLE
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The initial terms and their prime indices:
10: {1,3}
21: {2,4}
40: {1,1,1,3}
55: {3,5}
84: {1,1,2,4}
90: {1,2,2,3}
91: {4,6}
160: {1,1,1,1,1,3}
187: {5,7}
189: {2,2,2,4}
210: {1,2,3,4}
220: {1,1,3,5}
247: {6,8}
250: {1,3,3,3}
336: {1,1,1,1,2,4}
360: {1,1,1,2,2,3}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Select[Range[0, 100], ats[primeMS[#]]==-2&]
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CROSSREFS
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Below we use k to indicate alternating sum.
These are the positions of -2's in A316524.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344606 counts alternating permutations of prime indices.
Cf. A000037, A000070, A001791, A027187, A028260, A239830, A341446, A344609, A344616, A344651, A345197, A345910, A345912, A345961.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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