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A345960
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Numbers whose prime indices have alternating sum 2.
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7
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3, 12, 27, 30, 48, 70, 75, 108, 120, 147, 154, 192, 243, 270, 280, 286, 300, 363, 432, 442, 480, 507, 588, 616, 630, 646, 675, 750, 768, 867, 874, 972, 1080, 1083, 1120, 1144, 1200, 1323, 1334, 1386, 1452, 1470, 1587, 1728, 1750, 1768, 1798, 1875, 1920, 2028
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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 odd Omega (A001222) and exactly two odd conjugate prime indices. The version for even Omega is A345962, and the union is A345961. Conjugate prime indices are listed by A321650 and ranked by A122111.
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LINKS
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EXAMPLE
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The initial terms and their prime indices:
3: {2}
12: {1,1,2}
27: {2,2,2}
30: {1,2,3}
48: {1,1,1,1,2}
70: {1,3,4}
75: {2,3,3}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
147: {2,4,4}
154: {1,4,5}
192: {1,1,1,1,1,1,2}
243: {2,2,2,2,2}
270: {1,2,2,2,3}
280: {1,1,1,3,4}
286: {1,5,6}
300: {1,1,2,3,3}
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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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These partitions are counted by A000097.
These multisets are counted by A120452.
The version for reversed alternating sum is A345961.
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, A028260, A035363, A239830, A341446, A344609, A344610, A344651, A344741, A345197.
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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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