A007306 -id:A007306 - cp's OEIS frontend

A065674 Positions of the elements of the quasicyclic group Z+(2a+1)/(2^b) [a > 0 and a < 2^(b-1), b > 0] at the ]0,1[ side of the Stern-Brocot Tree (A007305/A007306).

1, 4, 7, 64, 10, 13, 127, 16384, 67, 79, 46, 49, 112, 124, 32767, 1073741824, 2050, 262, 139, 151, 2560, 352, 766, 769, 415, 3583, 232, 244, 505, 4093, 2147483647, 4611686018427387904, 4194307, 32776, 16447, 16639, 1057, 34816, 571, 583, 310

Offset: 1

Antti Karttunen, Nov 22 2001

nonn

			The fraction 1/2 is at the root (position 1), 1/4 is the left child of its left child, in the position 4 (when the tree is traversed in left-to-right, breadth-first fashion), while 3/4 is the right child of the right child of the root (pos. 7), 1/8 is at the position 64 (6 steps down the left branch from the root) and 3/8 is the right child of the left child of the root, at the position 10, etc.

Sean A. Irvine, Java program (github)
Index entries for sequences related to Stern's sequences

Permutation of A065810. Cf. A065658, A065675.

QuasiCyclics2_pos_in_0_1_SB_tree := proc(t) local num,den; den := 2^(1+floor_log_2(t)); num := (2*(t-(den/2)))+1; RETURN(frac2position_in_0_1_SB_tree(num/den)); end;
[seq(QuasiCyclics2_pos_in_0_1_SB_tree(j), j=1..128)]
# For missing Maple functions follow A065658.

A065936 a(n) is the integer (reduced squarefree) under the square root obtained when the inverse of a variant of Minkowski's question mark function is applied to the n-th ratio A007305(n+1)/A007306(n+1) in the left-hand subtree of Stern-Brocot tree and zero when it results a rational value.

Original entry on oeis.org

0, 5, 5, 0, 2, 2, 0, 2, 3, 0, 3, 3, 0, 3, 2, 5, 13, 17, 2, 17, 37, 5, 13, 13, 5, 37, 17, 2, 17, 13, 5, 3, 17, 3, 37, 21, 13, 10, 37, 3, 401, 6, 13, 10, 401, 0, 17, 17, 0, 401, 10, 13, 6, 401, 3, 37, 10, 13, 21, 37, 3, 17, 3, 0, 37, 10, 0, 401, 506, 17, 5, 401, 37, 21610, 730, 5, 1373

Offset: 1

Antti Karttunen, Dec 07 2001

nonn

Note: the underlying function N2Qv (see the Maple code) maps natural numbers 1, 2, 3, 4, 5, ..., through all the positive rationals in the open range (0,1): 1/2, 1/3, 2/3, 1/4, 2/5, 3/5, ... bijectively to the union of positive rationals and quadratic surds. A065937 gives similar mapping involving the inverse of the standard Minkowski's question mark function.

Note the symmetry of rows 0; 5,5; 0,2,2,0; 2,3,0,3,3,0,3,2; 5,13,17,2,17,37,5,13,13,5,37,17,2,17,13,5; ... emanating from the symmetry present in A007306.

			The first few values for this mapping are N2Qv(1) = 1, N2Qv(2) = (sqrt(5)-1)/2, N2Qv(3) = (sqrt(5)+1)/2, N2Qv(4) = 1/2, N2Qv(5) = sqrt(2)/2, N2Qv(6) = sqrt(2), N2Qv(7) = 2, N2Qv(8) = sqrt(2)-1

Eric Weisstein's World of Mathematics, Quadratic Irrational Number.
Index entries for sequences related to Minkowski's question mark function
Index entries for sequences related to Stern's sequences

a(n) = A065937(A065934(n)). Positions of the zeros are given by A065810. Positions of sqrt(n) in this mapping: A065938.

[seq(find_sqrt(N2Qv(j)),j=1..512)];
N2Qv := proc(n) local m; m := n + 2^floor_log_2(n); Inverse_of_Variant_of_MinkowskisQMark(A007305(m+1)/A047679(m-1)); end;
Inverse_of_Variant_of_MinkowskisQMark := proc(r) local x,y,b,d,k,s,i,q; x := numer(r); y := denom(r); if(y = 2*x) then RETURN(1); fi; b := []; d := []; k := 0; s := 0; i := 0; while(x <> 0) do q := floor(x/y); if(i > 0) then b := [op(b),q]; d := [op(d),x]; fi; x := 2*(x-(q*y)); if(member(x,d,'k') and (k > 1) and (b[k] <> b[k-1]) and (q <> floor(x/y))) then s := eval_periodic_confrac_tail(list2runcounts(b[k..nops(b)])); b := b[1..(k-1)]; break; fi; i := i+1; od; if(0 = k) then b := b[1..(nops(b)-1)]; b := [op(b),b[nops(b)]]; fi; if(r < (1/2)) then RETURN(factor(eval_confrac([0,op(list2runcounts(b))],s))); else RETURN(factor(eval_confrac(list2runcounts(b),s))); fi; end;
eval_confrac := proc(c,z) local x,i; x := z; for i in reverse(c) do x := (`if`((0=x),x,(1/x)))+i; od; RETURN(x); end;
eval_periodic_confrac_tail := proc(c) local x,i,u,r; x := (eval_confrac(c,u) - u) = 0; r := [solve(x,u)]; RETURN(max(r[1],r[2])); end;
list2runcounts := proc(b) local a,p,y,c; if(0 = nops(b)) then RETURN([]); fi; a := []; c := 0; p := b[1]; for y in b do if(y <> p) then a := [op(a),c]; c := 0; p := y; fi; c := c+1; od; RETURN([op(a),c]); end;
find_sqrt := proc(x) local n,i,y; n := nops(x); if(n < 2) then RETURN(0); fi; if((2 = n) and (`^` = op(0,x)) and (1/2 = op(2,x))) then RETURN(op(1,x)); else for i from 0 to n do y := find_sqrt(op(i,x)); if(y <> 0) then RETURN(y); fi; od; RETURN(0); fi; end;

Description clarified by Antti Karttunen, Aug 26 2006

A283973 Numbers n such that A007306(n) = A283986(n); positions of zeros in A283988.

Original entry on oeis.org

1, 3, 4, 9, 10, 12, 13, 15, 16, 21, 22, 27, 28, 33, 36, 37, 48, 49, 60, 61, 64, 78, 84, 85, 87, 88, 90, 91, 93, 94, 99, 100, 102, 103, 105, 106, 108, 109, 115, 129, 130, 133, 135, 136, 141, 144, 145, 153, 159, 160, 162, 171, 172, 189, 190, 192, 193, 195, 196, 213, 214, 223, 225, 226, 232, 240, 241, 244, 249, 250, 252, 255, 256

Offset: 1

Antti Karttunen, Mar 21 2017

Equally, numbers n for which A007306(n) = A283987(n), or equally, numbers n for which A283986(n) = A283987(n).

Numbers n such that the binary representations of A002487(n-1) and A002487(n) have no 1-bits in common shared positions.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A283974 (complement).

Cf. A002487, A007306, A283986, A283987, A283988.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Map[Function[n, If[EvenQ@ n, a[n/2], BitOr[a[#], a[# + 1]] &[(n - 1)/2]]], 2 Range[99] - 1] (* Michael De Vlieger, Mar 22 2017 *)

A(n) = if(n<2, n, if(n%2, A(n\2) + A((n + 1)/2), A(n/2)));
D(n) = if(n<1, 1, sum(k=0, n, binomial(n + k - 1, 2*k)%2)) /* A007306 */
for(n=1, 300, if(bitor(A(n - 1), A(n)) == D(n), print1(n,", "))) \\ Indranil Ghosh, Mar 23 2017

A049448 Sum of numerator and denominator of fractions in Farey tree A007305/A007306.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 7, 6, 9, 11, 10, 11, 13, 12, 9, 7, 11, 14, 13, 15, 18, 17, 13, 14, 19, 21, 18, 17, 19, 16, 11, 8, 13, 17, 16, 19, 23, 22, 17, 19, 26, 29, 25, 24, 27, 23, 16, 17, 25, 30, 27, 29, 34, 31, 23, 22, 29, 31, 26, 23, 25, 20, 13, 9, 15, 20, 19, 23

Offset: 0

N. J. A. Sloane, Gregory V. Richardson

J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

Subset of A007306. Cf. A049465-A049468.

A007305+A007306.

A065810 Sorted positions of the elements of the quasicyclic group Z+(2a+1)/(2^b) [a > 0 and a < 2^(b-1), b > 0] at the ]0,1[ side of the Stern-Brocot Tree (A007305/A007306).

Original entry on oeis.org

1, 4, 7, 10, 13, 46, 49, 64, 67, 79, 112, 124, 127, 139, 151, 232, 244, 262, 310, 325, 349, 352, 364, 403, 415, 418, 442, 457, 505, 571, 583, 661, 685, 766, 769, 850, 874, 952, 964, 1057, 1126, 1432, 1519, 1552, 1639, 1945, 2014, 2050, 2140, 2434, 2458

Offset: 1

Antti Karttunen, Nov 22 2001

nonn

It is easily proved that in the denominators given by A007306, the even values occur only at every third position, but can one find a simple rule for these positions of the denominators which are the powers of 2 only?

Index entries for sequences related to Stern's sequences

Permutation of A065674. Cf. A065811, A065812. Gives the positions of zeros in A065936.

A049449 Product of numerator and denominator of fractions in Farey tree A007305/A007306.

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 10, 15, 12, 5, 14, 24, 21, 28, 40, 35, 20, 6, 18, 33, 30, 44, 65, 60, 36, 45, 84, 104, 77, 70, 88, 63, 30, 7, 22, 42, 39, 60, 90, 85, 52, 70, 133, 168, 126, 119, 152, 112, 55, 66, 144, 209, 170, 198, 273, 228, 126, 117, 204, 234, 165, 130, 154, 99, 42, 8

Offset: 0

N. J. A. Sloane

Sum of reciprocals of each row (n > 1) equals 1/2. E.g., 1/3 + 1/6, 1/4 + 1/10 + 1/15 + 1/12 and so on. The link to cut-the-knot gives credit to Pierre Lamothe for this observation. - Joshua Zucker, May 11 2006

J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

Ivan Neretin, Table of n, a(n) for n = 0..4096
Alexander Bogomolny, Cut-the-knot on Stern-Brocot trees

See comment in A119272.

A007305*A007306.

More terms from Megan Wawro (s1095813(AT)cedarville.edu)

More terms from Joshua Zucker, May 11 2006

A057432 Obtained by reading first the numerator then the denominator of fractions in left-hand half of Stern-Brocot tree (A007305/A007306).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 2, 5, 3, 5, 3, 4, 1, 5, 2, 7, 3, 8, 3, 7, 4, 7, 5, 8, 5, 7, 4, 5, 1, 6, 2, 9, 3, 11, 3, 10, 4, 11, 5, 13, 5, 12, 4, 9, 5, 9, 7, 12, 8, 13, 7, 11, 7, 10, 8, 11, 7, 9, 5, 6, 1, 7, 2, 11, 3, 14, 3, 13, 4, 15, 5, 18, 5, 17, 4, 13, 5, 14, 7, 19, 8, 21, 7, 18, 7, 17, 8, 19, 7

Offset: 0

N. J. A. Sloane, Sep 08 2000

			The tree begins:
                                     1/1
                                     1/2
                  1/3                                   2/3
        1/4                 2/5               3/5                 3/4
    1/5      2/7       3/8       3/7     4/7       5/8       5/7      4/5
  1/6 2/9 3/11 3/10 4/11 5/13 5/12 4/9 5/9 7/12 8/13 7/11 7/10 8/11 7/9 5/6

N. J. A. Sloane, Stern-Brocot or Farey Tree
Index entries for sequences related to Stern's sequences

Cf. A007305, A047679, A007306, A002487, A057431.

Related to the Kepler tree A294442 via row permutations given by A088208 or A131271.

sbt[n_]:=Module[{P,L,Y},P={{1,0},{1,1}};L={{1,1},{0,1}};Y={{1,0},{0,1}}; w[b_]:=Fold[ #1.If[ #2==0,L,P]&,Y,b]; u[a_]:={a[[2,1]]+a[[2,2]],a[[1,1]]+a[[1,2]]}; s[l_]:={l,{Last[l],First[l]}}; Map[s,Map[u,Map[w,Part[Partition[Tuples[{0,1},n],2^(n-1)],1]]]]]
Flatten[Append[{1,1},Table[Map[First,sbt[i]],{i,1,6}]]] (* Peter Luschny, Apr 27 2009 *)

More terms from Alford Arnold, Sep 11 2000

More terms from Joshua Zucker, May 11 2006

A065675 The exponent of 2 in the fractions of the range ]0,1[ Stern-Brocot tree (A007305/A007306) [1/2, 1/3, 2/3, 1/4, 2/5, 3/5, 3/4, 1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, ...].

Original entry on oeis.org

-1, 0, 1, -2, 1, 0, -2, 0, 1, -3, 0, 2, -3, 0, 2, -1, 1, 0, -1, 2, 0, -2, 2, 0, -2, 3, 0, -1, 3, 0, -1, 0, 1, -1, 0, 2, -1, 0, 2, -1, 0, 3, -1, 0, 3, -4, 0, 1, -4, 0, 1, -1, 0, 2, -1, 0, 2, -1, 0, 1, -1, 0, 1, -3, 1, 0, -4, 2, 0, -1, 2, 0, -1, 3, 0, -3, 3, 0, -4, 1, 0, -1, 1, 0, -1, 2, 0, -1, 2, 0, -1, 1, 0, -2, 1, 0, -2, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0

Offset: 1

Antti Karttunen, Nov 22 2001

sign

The exponent is negative when the denominator (A007306) is even. These occur as every third term.

Index entries for sequences related to Stern's sequences

Cf. A065674, A065676, A065658.

[seq(exp_of_2(SternBrocot0_1frac(j)),j=1..128)];
SternBrocot0_1frac := proc(n) local m; m := n + 2^floor_log_2(n); SternBrocotTreeNum(m)/SternBrocotTreeDen(m); end;
exp_of_2 := proc(x) local f,m; f := ifactors(x)[2]; for m in f do if(2 = m[1]) then RETURN(m[2]); fi; od; RETURN(0); end;

A318316 Multiplicative with a(p^e) = 2^A007306(e).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 16, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 16, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 32, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 16, 8, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4, 32, 2, 8, 8, 16, 2, 8, 2, 16, 8

Offset: 1

Antti Karttunen, Aug 31 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A007306, A318307, A318322.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A007306(n) = if(!n,1,A002487(n+n-1));
A318316(n) = factorback(apply(e -> 2^A007306(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318316(n): return 1<Chai Wah Wu, May 18 2023

a(n) = 2^A318322(n).

a(n) = A318307(A003557(n^2)) = A318307(A003557(n))*A318307(n).

A318322 Additive with a(p^e) = A007306(e).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 4, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 5, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 5, 1, 3, 3, 4, 1, 3, 1, 4, 3

Offset: 1

Antti Karttunen, Aug 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A007306, A318306, A318316.

Differs from A122810 for the first time at n=48, where a(48) = 4, while A122810(48) = 5.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A007306(n) = if(!n,1,A002487(n+n-1));
A318322(n) = vecsum(apply(e -> A007306(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318322(n): return sum(sum(reduce(lambda x,y:(x[0],sum(x)) if int(y) else (sum(x),x[1]),bin((e<<1)-1)[-1:2:-1],(1,0))) for e in factorint(n).values()) # Chai Wah Wu, May 18 2023

a(n) = A007814(A318316(n)).

cp's OEIS Frontend

A065674 Positions of the elements of the quasicyclic group Z+(2a+1)/(2^b) [a > 0 and a < 2^(b-1), b > 0] at the ]0,1[ side of the Stern-Brocot Tree (A007305/A007306).

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Maple

A065936 a(n) is the integer (reduced squarefree) under the square root obtained when the inverse of a variant of Minkowski's question mark function is applied to the n-th ratio A007305(n+1)/A007306(n+1) in the left-hand subtree of Stern-Brocot tree and zero when it results a rational value.

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A283973 Numbers n such that A007306(n) = A283986(n); positions of zeros in A283988.

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A049448 Sum of numerator and denominator of fractions in Farey tree A007305/A007306.

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A065810 Sorted positions of the elements of the quasicyclic group Z+(2a+1)/(2^b) [a > 0 and a < 2^(b-1), b > 0] at the ]0,1[ side of the Stern-Brocot Tree (A007305/A007306).

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A049449 Product of numerator and denominator of fractions in Farey tree A007305/A007306.

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A057432 Obtained by reading first the numerator then the denominator of fractions in left-hand half of Stern-Brocot tree (A007305/A007306).

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A065675 The exponent of 2 in the fractions of the range ]0,1[ Stern-Brocot tree (A007305/A007306) [1/2, 1/3, 2/3, 1/4, 2/5, 3/5, 3/4, 1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, ...].

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A318316 Multiplicative with a(p^e) = 2^A007306(e).

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