cp's OEIS Frontend

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.

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A163236 Inverse permutation to A163235.

Original entry on oeis.org

0, 2, 1, 4, 9, 5, 13, 8, 6, 11, 3, 7, 24, 17, 18, 12, 35, 27, 43, 34, 14, 20, 19, 26, 62, 51, 52, 42, 32, 41, 25, 33, 28, 37, 21, 29, 58, 47, 48, 38, 10, 16, 15, 22, 31, 23, 39, 30, 112, 97, 98, 84, 70, 83, 59, 71, 73, 61, 85, 72, 40, 50, 49, 60, 135, 119, 151, 134, 90
Offset: 0

Views

Author

Antti Karttunen, Jul 29 2009

Keywords

Links

Crossrefs

Inverse: A163235. a(n) = A163234(A057300(n)). Cf. A163234.

Programs

  • Python
    def A(x, y): return (((x + y)**2) + x + 3*y)//2
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a059905(n): return sum([(n>>2*i&1)<Indranil Ghosh, Jun 25 2017
  • Scheme
    (define (A163236 n) (A001477bi (A006068 (A059906 n)) (A006068 (A059905 n))))
(define (A001477bi x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))

Formula

a(n) = A001477bi(A006068(A059906(n)),A006068(A059905(n))), where A001477bi(x,y) = (((x+y)^2)+x+(3y))/2.

A318703 For any n >= 0 with binary expansion Sum_{k=0..w} b_k * 2^k, let f(n) = Sum_{k=0..w} b_k * i^k * 2^floor(k/2) (where i denotes the imaginary unit); a(n) is the imaginary part of f(n).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, -2, -2, -1, -1, -2, -2, -1, -1, 0, 0, 1, 1, 0, 0, 1, 1, -2, -2, -1, -1, -2, -2, -1, -1, 4, 4, 5, 5, 4, 4, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 2, 2, 3, 3, 2, 2, 3, 3, 0, 0, 1, 1, 0, 0, 1, 1, -2, -2, -1, -1, -2, -2, -1
Offset: 0

Views

Author

Rémy Sigrist, Sep 01 2018

Keywords

Comments

See A318702 for the real part of f and additional comments.

Links

Crossrefs

Cf. A053985, A059906, A318702.

Programs

  • Mathematica
    Array[Im[Total@ MapIndexed[#1*I^(First@ #2 - 1)*2^Floor[(First@ #2 - 1)/2] &, Reverse@ IntegerDigits[#, 2]]] &, 75, 0] (* Michael De Vlieger, Sep 02 2018 *)
  • PARI
    a(n) = my (b=Vecrev(binary(n))); imag(sum(k=1, #b, b[k] * I^(k-1) * 2^floor((k-1)/2)))

Formula

a(n) = A053985(A059906(n)).
a(2*n) = a(2*n + 1) for any n >= 0.
a(4 * k) = -2 * a(k) for any k >= 0.

A371459 For any positive integer with binary digits (b_1, ..., b_w) (where b_1 = 1), the binary digits of a(n), possibly with leading zeros, are (b_2, b_4, ..., b_{floor(w/2) * 2}); a(0) = 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 3, 2, 3, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 0, 1, 0, 1, 2, 3, 2, 3, 0, 1, 0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 4, 5, 4, 5, 6, 7, 6, 7, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 0, 0, 1, 1, 0, 0, 1
Offset: 0

Views

Author

Rémy Sigrist, Mar 24 2024

Keywords

Comments

In other words, we keep even-indexed bits.
Every integer appears infinitely many times in the sequence.

Examples

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     0      10          0
   3     1      11          1
   4     0     100          0
   5     0     101          0
   6     1     110          1
   7     1     111          1
   8     0    1000          0
   9     1    1001          1
  10     0    1010          0
  11     1    1011          1
  12     2    1100         10
  13     3    1101         11
  14     2    1110         10
  15     3    1111         11
  16     0   10000          0

Links

Crossrefs

See A371442 for the sequence related to odd-indexed bits.
See A059906 and A063695 for similar sequences.
Cf. A000695, A001196, A126684.

Programs

  • Mathematica
    A371459[n_] := FromDigits[IntegerDigits[n, 2][[2;;-1;;2]], 2];
Array[A371459, 100, 0] (* Paolo Xausa, Mar 28 2024 *)
  • PARI
    a(n) = { my (b = binary(n)); fromdigits(vector(#b\2, k, b[2*k]), 2); }
  • Python
    def A371459(n): return int(bin(n)[3::2],2) if n>1 else 0 # Chai Wah Wu, Mar 27 2024

Formula

a(n) = 0 iff n belongs to A126684.
a(A000695(n)) = 0.
a(A001196(n)) = n.

A152819 "Upper primes" (see A152754).

Original entry on oeis.org

2, 11, 37, 41, 43, 47, 59, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 227, 229, 233, 239, 251, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727
Offset: 1

Views

Author

Vladimir Shevelev, Dec 13 2008

Keywords

Links

Crossrefs

Cf. A152754.

Programs

  • Mathematica
    fh[n_,h_] := If[h==1, Mod[n,2], If[Mod[n,4]>=2,1,0]]; half[n_, h_ ] := Module[{t=1, s=0, m=n}, While[m>0, s += fh[m,h]*t; m=Quotient[m,4]; t *= 2]; s]; mb[n_] := FromDigits[Riffle[IntegerDigits[n, 2], 0], 2]; aQ[n_] := PrimeQ[n] && mb[half[ n,1]] < mb[half[n, 2]]; Select[Range[730], aQ] (* Amiram Eldar, Dec 16 2018 from the PARI code *)
  • PARI
    a000695(n) = fromdigits(binary(n), 4);
half1(n) = { my(t=1, s=0); while(n>0, s += (n%2)*t; n \= 4; t *= 2); (s); }; \\ A059905
half2(n) = { my(t=1, s=0); while(n>0, s += ((n%4)>=2)*t; n \= 4; t *= 2); (s); }; \\ A059906
isok(n) = isprime(n) && a000695(half1(n)) < a000695(half2(n)); \\ Michel Marcus, Dec 15 2018

Extensions

More terms from Michel Marcus, Dec 15 2018

A296689 Let phi be the one-to-one mapping between binary trees and natural numbers described in the Tychonievich link. Let a(n) = min({phi^{-1}(t)| size(t)=n}); i.e., a(n) is the rank -- starting from 0 -- of the first tree the size of which is n.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 24, 30, 54, 64, 124, 244, 383, 503, 981, 1021, 1981, 3901, 6137, 8057, 13649, 16369, 32689, 65329, 98230, 130870, 229312, 261952, 491516, 524156, 1046388, 1048564, 2093044, 4182004, 8359924, 16715764, 25141220, 33497060, 58703812, 67059652, 125828996, 134184836, 259487492, 268435204, 536866564, 1073729284
Offset: 0

Views

Author

Philippe Esperet, Dec 18 2017

Keywords

Comments

Let v(n) = max({phi^{-1}(t)| size(t)=n}); v(n) is already known as A072639.
The interleaving process used by Tychonievich is not specific to base 2, each base b>=3 giving birth to a new a(n)-like sequence and a new v(n)-like sequence.
a(n) is the position of the first occurrence of n in A072644. - Andrey Zabolotskiy, Dec 20 2017
The tree-enumeration scheme of Tychonievich is similar, but not the same as "Recursive binary interleaving of binary trees" mentioned at my OEIS Wiki notes about Alternative Catalan Orderings. On the other hand, it seems to be the same (possibly up to the reflection of binary trees) as the ranking/unranking scheme mentioned in the section "Binary tree encoding with bijection" and in sequences A072634 - A072637 that are permutations of nonnegative integers induced by cross-ranking binary trees between such a "dense" binary interleaving ranking system and the standard lexicographic ordering of them (A014486). - Antti Karttunen, Dec 20 2017

Links

Crossrefs

Cf. A014486, A059905, A059906, A072639, A072634, A072635, A072644.

Programs

  • OCaml
    let rec evenOdd=function(*Luther Tychonievich decomposition*)
| n when n<=1 -> n,0
| n -> let ev,od=evenOdd(n/2) in
        2*od+n mod 2,ev
let rec cardImage=function
| n when n<=1 -> n
| n -> let ev,od=evenOdd(n-1) in 1+cardImage(ev)+cardImage(od)
let checkCatalanBis n=(*why 2*n+1 ? empirical...*)
  let (first,last)=(Array.make (2*n+1) 0,Array.make (2*n+1) 0) in
    for i=0 to 1 lsl n do
    let cai=cardImage i in
      last.(cai)<-1+last.(cai);
      if first.(cai)=0 then first.(cai)<-i done;
  (first,last)
  • Python
    def dei(n):
    n1 = n2 = 0
    bit = 1
    while n:
        if n&1:
            n1 += bit
        n >>= 1
        if n&1:
            n2 += bit
        n >>= 1
        bit <<= 1
    return (n1, n2)
r = [0]
for n in range(1, 100):
    r.append(1 + sum(r[x] for x in dei(n-1)))
print([r.index(x) for x in range(max(r)+1)])
# Andrey Zabolotskiy, Dec 20 2017

A147795 If n=A000695(k_n)+2*A000695(l_n), then a(n) is the number of nonnegative integers m

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 6, 6, 7, 7, 8, 8, 9, 9, 12, 12, 13, 13, 14, 14, 15, 15, 18, 18, 19, 19, 20, 20, 21, 21, 28, 28, 29, 29, 30, 30, 31, 31, 34, 34, 35, 35, 36, 36, 37, 37, 40, 40, 41, 41, 42, 42, 43, 43, 46, 46, 47, 47, 48, 48, 49, 49
Offset: 0

Views

Author

Vladimir Shevelev, Nov 13 2008

Keywords

Comments

Let us call integers m and n collinear one to another if either A059905(m)=A059905(n) or A059906(m)=A059906(n). Then a(n) is the number of noncollinear to n nonnegative integers not exceeding n.

Crossrefs

Cf. A000695, A054238, A059905, A059906.

Formula

Theorem: a(2n)=a(2n+1).

Extensions

More terms from Philippe Deléham, Oct 18 2011
Previous Showing 21-26 of 26 results.

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