A377337 Greatest positive integer k such that either k or k-1 is a multiple of A171977(n) but there does not exist any fully symmetric, k-celled, n-dimensional polyhypercube, or 0 if no such k exists.
0, 0, 77, 24
Offset: 1
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
HZeta := (s, v) -> Zeta(0, s, v): a := s -> (4^s*(HZeta(-s,1/4) - HZeta(-s,3/4)) + HZeta(-s,1)*(2^(s+1)-1))* 2^padic[ordp](2*(s+1),2): seq(a(n), n = 0..28);
a[n_] := 2^(IntegerExponent[n + 1, 2] + 1) (4^n (HurwitzZeta[-n, 1/4] - HurwitzZeta[-n, 3/4]) + HurwitzZeta[-n, 1] (2^(n + 1) - 1));
Table[FullSimplify[a[n]], {n, 0, 26}]
a(10^10) = 52010594272060810683. - _David A. Corneth_, Mar 26 2015
G := (x/((1-x)*(1+2*x))) * (1 + 2*x*mul(1+x^(2^k-1)+2*x^(2^k),k=0..20)); # N. J. A. Sloane, May 20 2009, Jun 05 2009 # From N. J. A. Sloane, Dec 25 2009: A139250 is T, A139251 is a. a:=[0,1,2,4]; T:=[0,1,3,7]; M:=10; for k from 1 to M do a:=[op(a),2^(k+1)]; T:=[op(T),T[nops(T)]+a[nops(a)]]; for j from 1 to 2^(k+1)-1 do a:=[op(a), 2*a[j+1]+a[j+2]]; T:=[op(T),T[nops(T)]+a[nops(a)]]; od: od: a; T;
CoefficientList[ Series[ (x/((1 - x)*(1 + 2x))) (1 + 2x*Product[1 + x^(2^k - 1) + 2*x^(2^k), {k, 0, 20}]), {x, 0, 53}], x] (* Robert G. Wilson v, Dec 06 2010 *)
a[0] = 0; a[n_] := a[n] = Module[{m, k}, m = 2^(Length[IntegerDigits[n, 2]] - 1); k = (2m^2+1)/3; If[n == m, k, k + 2 a[n - m] + a[n - m + 1] - 1]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 06 2018, after David A. Corneth *)
A139250(n,print_all=0)={my(p=[], /* set of "used" points. Points are written as complex numbers, c=x+iy. Toothpicks are of length 2 */ ee=[[0,1]], /* list of (exposed) endpoints. Exposed endpoints are listed as [c,d] where c=x+iy is the position of the endpoint, and d (unimodular) is the direction */ c,d,ne, cnt=1); print_all && print1("0,1"); n<2 && return(n); for(i=2,n, p=setunion(p, Set(Mat(ee~)[,1])); /* add endpoints (discard directions) from last move to "used" points */ ne=[]; /* new (exposed) endpoints */ for( k=1, #ee, /* add endpoints of new toothpicks if not among the used points */ setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne,Set([[c,d]])); setsearch(p, c-2*d) || ne=setunion(ne,Set([[c-2*d,-d]])); ); /* using Set() we have the points sorted, so it's easy to remove those which finally are not exposed because they touch a new toothpick */ forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] && k-- && ee=vecextract(ee,Str("^"k"..",k+1))); cnt+=#ee; /* each exposed endpoint will give a new toothpick */ print_all && print1(","cnt));cnt} \\ M. F. Hasler, Apr 14 2009
\\works for n > 0
a(n) = {my(k = (2*msb(n)^2 + 1) / 3); if(n==msb(n),k , k + 2*a(n-msb(n)) + a(n - msb(n) + 1) - 1)}
msb(n)=my(t=0);while(n>>t>0,t++);2^(t-1)\\ David A. Corneth, Mar 26 2015
def msb(n):
t = 0
while n>>t > 0:
t += 1
return 2**(t - 1)
def a(n):
k = (2 * msb(n)**2 + 1) / 3
return 0 if n == 0 else k if n == msb(n) else k + 2*a(n - msb(n)) + a(n - msb(n) + 1) - 1
[a(n) for n in range(101)] # Indranil Ghosh, Jul 01 2017, after David A. Corneth's PARI script
For example, 2^1|2, 2^2|4, 2^1|6, 2^3|8, 2^1|10, 2^2|12, ... giving the initial terms 1, 2, 1, 3, 1, 2, ...
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
1;
2,1;
3,1,2,1;
4,1,2,1,3,1,2,1;
5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
6,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
7,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,...
(End)
S(0) = {} S(1) = 1 S(2) = 1, 2, 1 S(3) = 1, 2, 1, 3, 1, 2, 1 S(4) = 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1. - Yann David (yann_david(AT)hotmail.com), Mar 21 2010
From _Joerg Arndt_, Nov 12 2012: (Start)
The 16 compositions of 5 as lists in lexicographic order:
[ n] a(n) composition
[ 1] [ 1] [ 1 1 1 1 1 ]
[ 2] [ 2] [ 1 1 1 2 ]
[ 3] [ 1] [ 1 1 2 1 ]
[ 4] [ 3] [ 1 1 3 ]
[ 5] [ 1] [ 1 2 1 1 ]
[ 6] [ 2] [ 1 2 2 ]
[ 7] [ 1] [ 1 3 1 ]
[ 8] [ 4] [ 1 4 ]
[ 9] [ 1] [ 2 1 1 1 ]
[10] [ 2] [ 2 1 2 ]
[11] [ 1] [ 2 2 1 ]
[12] [ 3] [ 2 3 ]
[13] [ 1] [ 3 1 1 ]
[14] [ 2] [ 3 2 ]
[15] [ 1] [ 4 1 ]
[16] [ 5] [ 5 ]
a(n) is the last part in each list.
(End)
From _Omar E. Pol_, Aug 20 2013: (Start)
Also written as a triangle in which the right border gives A000027 and row lengths give A011782 and row sums give A000079 the sequence begins:
1;
2;
1,3;
1,2,1,4;
1,2,1,3,1,2,1,5;
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6;
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7;
(End)
G.f. = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 2*x^6 + x^7 + 4*x^8 + x^9 + 2*x^10 + ...
a001511 n = length $ takeWhile ((== 0) . (mod n)) a000079_list -- Reinhard Zumkeller, Sep 27 2011
a001511 n | odd n = 1 | otherwise = 1 + a001511 (n `div` 2) -- Walt Rorie-Baety, Mar 22 2013
nmax=5;r=1;for n=2:nmax;r=[r n r];end % Adriano Caroli, Feb 26 2016
[Valuation(2*n,2): n in [1..105]]; // Bruno Berselli, Nov 23 2015
A001511 := n->2-wt(n)+wt(n-1); # where wt is defined in A000120 # This is the binary logarithm of the denominator of (256^n-1)B_{8n}/n, in Maple parlance a := n -> log[2](denom((256^n-1)*bernoulli(8*n)/n)). - Peter Luschny, May 31 2009 A001511 := n -> padic[ordp](2*n,2): seq(A001511(n), n=1..105); # Peter Luschny, Nov 26 2010 a:= n-> ilog2((Bits[Xor](2*n, 2*n-1)+1)/2): seq(a(n), n=1..50); # Gary Detlefs, Dec 13 2018
Array[ If[ Mod[ #, 2] == 0, FactorInteger[ # ][[1, 2]], 0] &, 105] + 1 (* or *)
Nest[ Flatten[ # /. a_Integer -> {1, a + 1}] &, {1}, 7] (* Robert G. Wilson v, Mar 04 2005 *)
IntegerExponent[2*n, 2] (* Alexander R. Povolotsky, Aug 19 2011 *)
myHammingDistance[n_, m_] := Module[{g = Max[m, n], h = Min[m, n]}, b1 = IntegerDigits[g, 2]; b2 = IntegerDigits[h, 2, Length[b1]]; HammingDistance[b1, b2]] (* Vladimir Shevelev A206853 *) Table[ myHammingDistance[n, n - 1], {n, 111}] (* Robert G. Wilson v, Apr 05 2012 *)
Table[Position[Reverse[IntegerDigits[n,2]],1,1,1],{n,110}]//Flatten (* Harvey P. Dale, Aug 18 2017 *)
a(n) = sum(k=0,floor(log(n)/log(2)),floor(n/2^k)-floor((n-1)/2^k)) /* Ralf Stephan */
a(n)=if(n%2,1,factor(n)[1,2]+1) /* Jon Perry, Jun 06 2004 */
{a(n) = if( n, valuation(n, 2) + 1, 0)}; /* Michael Somos, Sep 30 2006 */
{a(n)=if(n==1,1,polcoeff(x-sum(k=1, n-1, a(k)*x^k*(1-x^k)*(1-x+x*O(x^n))), n))} /* Paul D. Hanna, Jun 22 2007 */
def a(n): return bin(n)[2:][::-1].index("1") + 1 # Indranil Ghosh, May 11 2017
A001511 = lambda n: (n&-n).bit_length() # M. F. Hasler, Apr 09 2020
def A001511(n): return (~n & n-1).bit_length()+1 # Chai Wah Wu, Jul 01 2022
[valuation(2*n,2) for n in (1..105)] # Bruno Berselli, Nov 23 2015
(define (A001511 n) (let loop ((n n) (e 1)) (if (odd? n) e (loop (/ n 2) (+ 1 e))))) ;; Antti Karttunen, Oct 06 2017
2^3 divides 24, but 2^4 does not divide 24, so a(24) = 8. 2^0 divides 25, but 2^1 does not divide 25, so a(25) = 1. 2^1 divides 26, but 2^2 does not divide 26, so a(26) = 2. Per _Marc LeBrun_'s 2000 comment, a(n) can also be determined with bitwise operations in two's complement. For example, given n = 48, we see that n in binary in an 8-bit byte is 00110000 while -n is 11010000. Then 00110000 AND 11010000 = 00010000, which is 16 in decimal, and therefore a(48) = 16. G.f. = x + 2*x^2 + x^3 + 4*x^4 + x^5 + 2*x^6 + x^7 + 8*x^8 + x^9 + ...
import Data.Bits ((.&.)) a006519 n = n .&. (-n) :: Integer -- Reinhard Zumkeller, Mar 11 2012, Dec 29 2011
using IntegerSequences [EvenPart(n) for n in 1:102] |> println # Peter Luschny, Sep 25 2021
[2^Valuation(n, 2): n in [1..100]]; // Vincenzo Librandi, Mar 27 2015
with(numtheory): for n from 1 to 200 do if n mod 2 = 1 then printf(`%d,`,1) else printf(`%d,`,2^ifactors(n)[2][1][2]) fi; od: A006519 := proc(n) if type(n,'odd') then 1 ; else for f in ifactors(n)[2] do if op(1,f) = 2 then return 2^op(2,f) ; end if; end do: end if; end proc: # R. J. Mathar, Oct 25 2010 A006519 := n -> 2^padic[ordp](n,2): # Peter Luschny, Nov 26 2010
lowestOneBit[n_] := Block[{k = 0}, While[Mod[n, 2^k] == 0, k++]; 2^(k - 1)]; Table[lowestOneBit[n], {n, 102}] (* Robert G. Wilson v Nov 17 2004 *)
Table[2^IntegerExponent[n, 2], {n, 128}] (* Jean-François Alcover, Feb 10 2012 *)
Table[BitAnd[BitNot[i - 1], i], {i, 1, 102}] (* Peter Luschny, Oct 10 2019 *)
{a(n) = 2^valuation(n, 2)};
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