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A005941
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Inverse of the Doudna sequence A005940.
(Formerly M0510)
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34
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1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 17, 12, 33, 18, 11, 16, 65, 14, 129, 20, 19, 34, 257, 24, 13, 66, 15, 36, 513, 22, 1025, 32, 35, 130, 21, 28, 2049, 258, 67, 40, 4097, 38, 8193, 68, 23, 514, 16385, 48, 25, 26, 131, 132, 32769, 30, 37, 72, 259, 1026, 65537, 44, 131073, 2050, 39, 64
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OFFSET
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1,2
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COMMENTS
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Question: Is there a simple proof that a(c) = c would never allow an odd composite c as a solution? See also A364551. - Antti Karttunen, Jul 30 2023
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REFERENCES
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J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = h(g(n,1,1), 0) / 2 + 1 with h(n, m) = if n=0 then m else h(floor(n/2), 2*m + n mod 2) and g(n, i, x) = if n=1 then x else (if n mod prime(i) = 0 then g(n/prime(i), i, 2*x+1) else g(n, i+1, 2*x). - Reinhard Zumkeller, Aug 23 2006
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MAPLE
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local k ;
for k from 1 do
if A005940(k) = n then # code reuse
return k;
end if;
end do ;
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MATHEMATICA
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f[n_] := Block[{p = Partition[ Split[ Join[ IntegerDigits[n - 1, 2], {2}]], 2]}, Times @@ Flatten[ Table[q = Take[p, -i]; Prime[ Count[ Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}] ]]; t = Table[ f[n], {n, 10^5}]; Flatten[ Table[ Position[t, n, 1, 1], {n, 64}]] (* Robert G. Wilson v, Feb 22 2005 *)
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PROG
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(Python)
from sympy import primepi, factorint
def A005941(n): return sum((1<<primepi(p)-1)<<i for i, p in enumerate(factorint(n, multiple=True)))+1 # Chai Wah Wu, Mar 11 2023
(PARI) A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552) - Antti Karttunen, Jul 30 2023
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CROSSREFS
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Cf. A029747 [known positions where a(n) = n], A364560 [where a(n) <= n], A364561 [where a(n) <= n and n is odd], A364562 [where a(n) > n], A364548 [where n divides a(n)], A364549 [where odd n divides a(n)], A364550 [where a(n) divides n], A364551 [where a(n) divides n and n is odd].
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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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