Search: a364550 -id:a364550
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A029747
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Numbers of the form 2^k times 1, 3 or 5.
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+10
50
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1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 4096, 5120, 6144, 8192, 10240, 12288, 16384, 20480, 24576, 32768, 40960, 49152, 65536, 81920, 98304, 131072, 163840, 196608
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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EXAMPLE
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128 = 2^7 * 1 is in the sequence as well as 160 = 2^5 * 5. - David A. Corneth, Sep 18 2020
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MATHEMATICA
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m = 200000; Select[Union @ Flatten @ Outer[Times, {1, 3, 5}, 2^Range[0, Floor[Log2[m]]]], # < m &] (* Amiram Eldar, Oct 15 2020 *)
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PROG
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CROSSREFS
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Even terms form a subsequence of A320674.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A005941
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Inverse of the Doudna sequence A005940.
(Formerly M0510)
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+10
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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1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 55, 60, 63, 64, 70, 72, 75, 77, 80, 81, 84, 90, 91, 96, 98, 99, 100, 105, 108, 110, 120, 121, 125, 126, 128, 135, 140, 143, 144, 147, 150, 154, 160, 162, 165, 168, 169, 175, 180, 182, 187, 189, 192, 195, 196
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OFFSET
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1,2
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COMMENTS
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Numbers k such that A005941(k) <= k.
If k is a term, then also 2*k is present in this sequence, and vice versa.
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LINKS
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PROG
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(PARI)
A156552(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]); res };
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CROSSREFS
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Positions of nonpositive terms in A364559.
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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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2, 3, 4, 6, 8, 12, 16, 24, 27, 32, 48, 54, 64, 96, 108, 128, 192, 216, 256, 315, 384, 432, 512, 630, 768, 864, 1024, 1260, 1536, 1728, 2048, 2520, 3003, 3072, 3456, 4096, 5040, 6006, 6144, 6912, 8192, 10080, 12012, 12288, 13824, 16384, 20160, 24024, 24576, 27648, 32768, 40320, 42757, 48048, 49152, 55296, 65536
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OFFSET
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1,1
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COMMENTS
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For k > 1, if k is a term, then also 2*k is present in this sequence, and vice versa.
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LINKS
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PROG
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(PARI)
A243071(n) = if(n<=2, n-1, 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]); ((3<<#binary(res\2))-res-1)); \\ (Combining programs given in A156552 and A054429)
isA364964(n) = ((n>1)&&!(n%A243071(n)));
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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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