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A349159
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Numbers whose sum of prime indices is twice their alternating sum.
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7
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1, 12, 63, 66, 112, 190, 255, 325, 408, 434, 468, 609, 805, 832, 931, 946, 1160, 1242, 1353, 1380, 1534, 1539, 1900, 2035, 2067, 2208, 2296, 2387, 2414, 2736, 3055, 3108, 3154, 3330, 3417, 3509, 3913, 4185, 4340, 4503, 4646, 4650, 4664, 4864, 5185, 5684, 5863
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OFFSET
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1,2
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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.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their alternating sum.
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LINKS
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FORMULA
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EXAMPLE
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The terms and their prime indices begin:
1: ()
12: (2,1,1)
63: (4,2,2)
66: (5,2,1)
112: (4,1,1,1,1)
190: (8,3,1)
255: (7,3,2)
325: (6,3,3)
408: (7,2,1,1,1)
434: (11,4,1)
468: (6,2,2,1,1)
609: (10,4,2)
805: (9,4,3)
832: (6,1,1,1,1,1,1)
931: (8,4,4)
946: (14,5,1)
1160: (10,3,1,1,1)
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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[1000], Total[primeMS[#]]==2*ats[primeMS[#]]&]
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CROSSREFS
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These partitions are counted by A000712 up to 0's.
A025047 counts alternating or wiggly compositions, complement A345192.
A116406 counts compositions with alternating sum >= 0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Cf. A000070, A000290, A001700, A028260, A045931, A120452, A195017, A241638, A257991, A257992, A325698, A345958, A349155.
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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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