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A357975
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Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.
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10
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1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 5, 3, 4, 1, 5, 4, 7, 2, 6, 3, 7, 2, 4, 5, 8, 3, 11, 4, 11, 1, 6, 5, 6, 4, 13, 7, 10, 2, 13, 6, 17, 3, 8, 7, 17, 2, 9, 4, 10, 5, 19, 8, 6, 3, 14, 11, 19, 4, 23, 11, 12, 1, 10, 6, 23, 5, 14, 6, 29, 4, 29, 13, 8, 7, 9, 10, 31
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OFFSET
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1,3
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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.
Also the Heinz number of the part-wise half (rounded down) of the partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Each n appears A000005(n) times at odd positions (infinitely many at even). To see this, note that our transformation does not distinguish between A066207 and A066208.
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LINKS
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FORMULA
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Completely multiplicative with a(prime(2k)) = prime(k) and a(prime(2k+1)) = prime(k). Cf. A297002.
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EXAMPLE
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The prime indices of n = 1501500 are {1,1,2,3,3,3,4,5,6}, so the prime indices of a(n) are {1,1,1,1,2,2,3}; hence we have a(1501500) = 720.
The 6 odd positions of 2124 are: 63, 99, 105, 165, 175, 275, with prime indices:
63: {2,2,4}
99: {2,2,5}
105: {2,3,4}
165: {2,3,5}
175: {3,3,4}
275: {3,3,5}
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MATHEMATICA
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Table[Times@@(If[#1<=2, 1, Prime[Floor[PrimePi[#1]/2]]^#2]&@@@FactorInteger[n]), {n, 100}]
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CROSSREFS
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Positions of 2's are 3 and A164095.
A004526 is floor(n/2), with an extra first zero.
A109763 lists primes of index floor(n/2).
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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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