Displaying 1-4 of 4 results found.
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Numbers k such that k is a multiple of A243071(k).
+0
2
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
COMMENTS
For k > 1, if k is a term, then also 2*k is present in this sequence, and vice versa.
PROG
(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)));
Numbers k for which A156552(k) < k.
+0
8
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
COMMENTS
Numbers k such that A005941(k) <= k.
If k is a term, then also 2*k is present in this sequence, and vice versa.
PROG
(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 };
CROSSREFS
Positions of nonpositive terms in A364559.
Numbers of the form 2^k times 1, 3 or 5.
+0
50
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
EXAMPLE
128 = 2^7 * 1 is in the sequence as well as 160 = 2^5 * 5. - David A. Corneth, Sep 18 2020
MATHEMATICA
m = 200000; Select[Union @ Flatten @ Outer[Times, {1, 3, 5}, 2^Range[0, Floor[Log2[m]]]], # < m &] (* Amiram Eldar, Oct 15 2020 *)
CROSSREFS
Even terms form a subsequence of A320674.
Inverse of the Doudna sequence A005940.
(Formerly M0510)
+0
34
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
COMMENTS
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
REFERENCES
J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
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
MAPLE
local k ;
for k from 1 do
if A005940(k) = n then # code reuse
return k;
end if;
end do ;
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
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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