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0, 1, 3, 2, 15, 6, 7, 4, 5, 30, 63, 12, 255, 14, 29, 8, 511, 10, 1023, 60, 13, 126, 2047, 24, 23, 510, 9, 28, 4095, 58, 31, 16, 125, 1022, 27, 20, 16383, 2046, 509, 120, 32767, 26, 65535, 252, 57, 4094, 262143, 48, 11, 46, 1021, 1020, 1048575, 18, 119, 56, 2045, 8190, 2097151, 116, 4194303, 62, 25, 32, 503, 250, 8388607, 2044, 4093, 54, 16777215, 40
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OFFSET
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1,3
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COMMENTS
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Any Mersenne prime (A000668) times any power of 2 (i.e., 2^k, for k>=0) is fixed by this sequence, including also all even perfect numbers.
This is a "tuned variant" of A243071, and has many of the same properties.
For example, for n > 1, A007814(a(n)) = A007814(n) - A209229(n), that is, this map preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is decremented by one, and in particular, a(2^k * n) = 2^k * a(n) for all n > 1. Also, like A243071, this bijection maps primes to the terms of A000225 (binary repunits). However, the "tuning" (A332213) has a specific effect that each Mersenne prime (A000668) is mapped to itself. Therefore the terms of A335431 are fixed by this map. Furthermore, I conjecture that there are no other fixed points. For the starters, see the proof in A335879, which shows that at least none of the terms of A335882 are fixed.
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LINKS
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FORMULA
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PROG
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CROSSREFS
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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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