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A308193
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Indices of records in A308190.
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4
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5, 6, 7, 10, 16, 17, 29, 53, 101, 197, 389, 773, 1542, 3079, 6154, 12304, 24604, 36901, 73798, 147592, 295180, 295517, 591030, 1182056, 1574849, 3149694, 4728211, 6299383, 12598762, 25197520, 25197533, 50395062, 100790119, 201580234, 403160464, 806320924, 1232145821, 2464291638
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OFFSET
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1,1
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COMMENTS
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For terms a(1) through a(16), with one exception, 2*a(n) - a(n+1) is either 4 or 5. Does this pattern continue, and if so, why?
The pattern does not continue. a(17) = 24604, a(18) = 36901.
Theorem:
1. All terms are even or prime.
2. If a(n+1) is even, then 2*a(n)-a(n+1) = 4.
3. a(n+1) <= 2*(a(n)-2).
Proof: If a(n+1) = x is even, then A111234(x) = 2+x/2 = y. If we assume that x >= 6, then y < x. Thus A308190(x) = A308190(y)+1, i.e., a(n) <= y. If a(n) < y, then A308190(2*(a(n)-2)) = A308190(a(n)) + 1.
Since a(n) is a record value, this means that the next record value is at most at 2*(a(n)-2), i.e., 2*(a(n)-2) < x = a(n+1), a contradiction.
Thus we have shown that if a(n+1) is even, then 2*a(n) = a(n+1)+4.
If a(n+1) = x is an odd composite with smallest prime factor p > 2, then A308190(x) = A308190(y)+1 where y = p+x/p. On the other hand, A308190(2*(y-2)) = A308190(y)+1. Since 2*(y-2) < x, this contradicts the fact that a(n+1) = x is a record value.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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