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A348617
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Numbers whose sum of prime indices is twice their negated alternating sum.
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4
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1, 10, 39, 88, 115, 228, 259, 306, 517, 544, 620, 783, 793, 870, 1150, 1204, 1241, 1392, 1656, 1691, 1722, 1845, 2369, 2590, 2596, 2775, 2944, 3038, 3277, 3280, 3339, 3498, 3692, 3996, 4247, 4440, 4935, 5022, 5170, 5226, 5587, 5644, 5875, 5936, 6200, 6321
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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 negated 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: ()
10: (3,1)
39: (6,2)
88: (5,1,1,1)
115: (9,3)
228: (8,2,1,1)
259: (12,4)
306: (7,2,2,1)
517: (15,5)
544: (7,1,1,1,1,1)
620: (11,3,1,1)
783: (10,2,2,2)
793: (18,6)
870: (10,3,2,1)
1150: (9,3,3,1)
1204: (14,4,1,1)
1241: (21,7)
1392: (10,2,1,1,1,1)
1656: (9,2,2,1,1,1)
1691: (24,8)
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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 A001523 up to 0's.
The reverse nonnegative version is A349160, counted by A006330 up to 0's.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Cf. A000984, A001700, A028260, A045931, A120452, A195017, A257991, A257992, A262977, A325698, A344619, A345958.
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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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