A260273
Successively add the smallest nonzero binary number that is not a substring.
Original entry on oeis.org
1, 3, 5, 8, 11, 15, 17, 20, 23, 27, 31, 33, 36, 39, 44, 51, 56, 61, 65, 68, 71, 76, 81, 84, 87, 91, 95, 99, 104, 111, 115, 120, 125, 129, 132, 135, 140, 145, 148, 151, 157, 165, 168, 171, 175, 179, 186, 190, 194, 199, 204, 209, 216, 223, 227, 232, 241, 246
Offset: 1
Begin with a(1)=1, in binary, "1". This contains the string "1" but not "10", so we add 2. Thus a(2)=1+2=3. This also contains "1" but not "10", so we move to a(3)=3+2=5. This contains "1" and "10" but not "11", so we add 3. Thus a(4)=5+3=8. (See A261018 for the successive numbers that are added. - _N. J. A. Sloane_, Aug 17 2015)
See also
A261396 (when a(n) just passes a power of 2),
A261416 (the limiting behavior just past a power of 2).
-
a260273 n = a260273_list !! (n-1)
a260273_list = iterate (\x -> x + a261461 x) 1
-- Reinhard Zumkeller, Aug 30 2015, Aug 17 2015
-
public static void main(String[] args) {
int a=1;
for(int iter=0;iter<100;iter++){
System.out.print(a+", ");
int inc;
for(inc=1; contains(a,inc); inc++);
a+=inc;
}
}
static boolean contains(int a,int test){
int mask=(Integer.highestOneBit(test)<<1)-1;
while(a >= test){
if((a & mask) == test) return true;
a >>= 1;
}
return false;
}
-
sublistQ[L1_, L2_] := Module[{l1 = Length[L1], l2 = Length[L2], k}, If[l2 <= l1, For[k = 1, k <= l1 - l2 + 1, k++, If[L1[[k ;; k + l2 - 1]] == L2, Return[True]]]]; False];
a[1] = 1; a[n_] := a[n] = Module[{bb = IntegerDigits[a[n-1], 2], k}, For[k = 1, sublistQ[bb, IntegerDigits[k, 2]], k++]; a[n-1] + k]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Apr 01 2016 *)
NestList[Function[k, k + FromDigits[#, 2] &@ SelectFirst[IntegerDigits[Range[2^8], 2], Length@ SequencePosition[IntegerDigits[k, 2], #] == 0 &]], 1, 64] (* Michael De Vlieger, Apr 01 2016, Version 10.1 *)
-
A260273_list, a = [1], 1
for i in range(10**3):
b, s = 1, format(a,'b')
while format(b,'b') in s:
b += 1
a += b
s = format(a,'b')
A260273_list.append(a) # Chai Wah Wu, Aug 26 2015
A261416
Let b(k) denote A260273(k). It appears that for k >= 200, whenever b(k) just passes a power of 2, 2^m say, the successive differences b(k)-2^m converge to this sequence.
Original entry on oeis.org
2, 5, 8, 11, 17, 20, 23, 29, 38, 43, 49, 54, 61, 70, 75, 81, 84, 87, 93, 102, 107, 114, 119, 128, 131, 136, 139, 145, 148, 151, 157, 167, 173, 180, 187, 196, 201, 206, 211, 218, 225, 230, 235, 244, 253, 262, 267, 273, 276, 279, 285, 294, 299, 305, 310, 317, 327, 333, 340, 343, 349, 358, 365, 372, 381
Offset: 0
At k=200, b(k)=b(200)=1026 has just passed 2^10. The successive differences b(200+i)-2^10 (i>=0) beyond this point are 2, 5, 8, 11, 17, 20, 23, 29, 38, 43, 49, 54, 61, 70, 75, 81, 84, 87, 93, 102, 107, 114, 119, 128, 131, 136, 139, 145, 148, 151, 157, [165, ...], which are the first 31 terms of the present sequence.
At k=371, b(371)=2050, and the successive differences b(371+i)-2^11 are 2, 5, ..., 279, 285, ... giving the first 51 terms of the present sequence.
A261644
Distance of A260273(n) to next power of 2.
Original entry on oeis.org
1, 1, 3, 8, 5, 1, 15, 12, 9, 5, 1, 31, 28, 25, 20, 13, 8, 3, 63, 60, 57, 52, 47, 44, 41, 37, 33, 29, 24, 17, 13, 8, 3, 127, 124, 121, 116, 111, 108, 105, 99, 91, 88, 85, 81, 77, 70, 66, 62, 57, 52, 47, 40, 33, 29, 24, 15, 10, 6, 2, 254, 251, 248, 245, 239
Offset: 1
. 1: 1
. 2: 1
. 3: 3
. 4: 8,5,1
. 5: 15,12,9,5,1
. 6: 31,28,25,20,13,8,3
. 7: 63,60,57,52,47,44,41,37,33,29,24,17,13,8,3
. 8: 127,124,121,116,111,108,105,99,91,88,85,81,77,70,... (27 terms)
. 9: 254,251,248,245,239,236,233,227,218,213,207,202,195,,... (49 terms)
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a261644 n = a261644_list !! (n-1)
a261644_list = zipWith (-)
(map a062383 a260273_list) $ map fromIntegral a260273_list
a261644_tabf = [1] : f (tail $ zip a261645_list a261644_list) where
f dxs = (map snd (dxs'' ++ [dx])) : f dxs' where
(dxs'', dx:dxs') = span ((<= 0) . fst) dxs
a261644_row n = a261644_tabf !! (n-1)
Original entry on oeis.org
1, 1, 1, 3, 5, 7, 15, 27, 49, 90, 171, 326, 613, 1174, 2255, 4333, 8386, 16213, 31457, 61097, 118577, 230322, 447767, 870570, 1693887, 3297343, 6423347, 12523562, 24435171, 47707012, 93215393, 182260604, 356622869, 698284551, 1368195878, 2682527954, 5262729874, 10330704931
Offset: 1
A261788
a(n) is the smallest k such that A261786(k) >= 3^n.
Original entry on oeis.org
1, 2, 5, 12, 30, 81, 224, 626, 1747, 4909, 13811, 38934, 109889, 310666, 880125, 2500221, 7125406, 20376598, 58472481, 168349612, 486198698, 1408140693, 4088769215, 11899761717, 34703682407
Offset: 0
-
a261788 n = a261788_list !! (n-1)
a261788_list = f 1 1 a261786_list' where
f z k (x:xs) | x >= z = k : f (3 * z) (k + 1) xs
| otherwise = f z (k + 1) xs
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ts(n) = Str(fromdigits(digits(n, 3))); \\ A007089
f(n) = my(s=ts(n), k=1); while (#strsplit(s, ts(k)) != 1, k++); k; \\ A261787
lista(nn) = my(last=1, k=0, p=3^k); for (n=1, nn, if (last >= p, print1(n, ", "); k++; p = 3^k); last += f(last);); \\ Michel Marcus, Feb 06 2022
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