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A345961
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Numbers whose prime indices have reverse-alternating sum 2.
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10
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3, 10, 12, 21, 27, 30, 40, 48, 55, 70, 75, 84, 90, 91, 108, 120, 147, 154, 160, 187, 189, 192, 210, 220, 243, 247, 250, 270, 280, 286, 300, 336, 360, 363, 364, 391, 432, 442, 462, 480, 490, 495, 507, 525, 551, 588, 616, 630, 640, 646, 675, 713, 748, 750, 756
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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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.
Also numbers with exactly two odd conjugate prime indices. The restriction to odd omega is A345960, and the restriction to even omega is A345962.
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LINKS
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EXAMPLE
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The initial terms and their prime indices:
3: {2}
10: {1,3}
12: {1,1,2}
21: {2,4}
27: {2,2,2}
30: {1,2,3}
40: {1,1,1,3}
48: {1,1,1,1,2}
55: {3,5}
70: {1,3,4}
75: {2,3,3}
84: {1,1,2,4}
90: {1,2,2,3}
91: {4,6}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Select[Range[100], sats[primeMS[#]]==2&]
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CROSSREFS
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Below we use k to indicate reverse-alternating sum.
These multisets are counted by A000097.
These are the positions of 2's in A344616.
A000070 counts partitions with alternating sum 1.
A027187 counts partitions with reverse-alternating sum <= 0.
A088218 also counts compositions with alternating sum 0, ranked by A344619.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices.
A344606 counts alternating permutations of prime indices.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000984, A001791, A025047, A027193, A239830, A341446, A344611, A344650, A344651, A344743, A345910, A345911, A345918.
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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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