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A345959
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Numbers whose prime indices have alternating sum -1.
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10
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6, 15, 24, 35, 54, 60, 77, 96, 135, 140, 143, 150, 216, 221, 240, 294, 308, 315, 323, 375, 384, 437, 486, 540, 560, 572, 600, 667, 693, 726, 735, 864, 875, 884, 899, 960, 1014, 1147, 1176, 1215, 1232, 1260, 1287, 1292, 1350, 1500, 1517, 1536, 1715, 1734, 1748
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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 one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.
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LINKS
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EXAMPLE
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The initial terms and their prime indices:
6: {1,2}
15: {2,3}
24: {1,1,1,2}
35: {3,4}
54: {1,2,2,2}
60: {1,1,2,3}
77: {4,5}
96: {1,1,1,1,1,2}
135: {2,2,2,3}
140: {1,1,3,4}
143: {5,6}
150: {1,2,3,3}
216: {1,1,1,2,2,2}
221: {6,7}
240: {1,1,1,1,2,3}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Select[Range[0, 100], ats[primeMS[#]]==-1&]
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CROSSREFS
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These multisets are counted by A000070.
These are the positions of -1's in A316524.
A027187 counts partitions with reverse-alternating sum <= 0.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344607 counts partitions with reverse-alternating sum >= 0.
A344616 gives the alternating sum of reversed prime indices.
Cf. A000097, A028260, A035363, A236913, A239830, A341446, A344609, A344610, A344651, A345919, 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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