A002487 -id:A002487 - cp's OEIS frontend

A284459 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A002487/A002487' (Calkin-Wilf) into the enumeration system A245327/A245328, and A162911/A162912 (Drib) into A020651/A020650 (Yu-Ting inverted).

1, 2, 3, 6, 5, 4, 7, 10, 13, 12, 11, 14, 9, 8, 15, 26, 21, 20, 27, 22, 25, 24, 23, 18, 29, 28, 19, 30, 17, 16, 31, 42, 53, 52, 43, 54, 41, 40, 55, 50, 45, 44, 51, 46, 49, 48, 47, 58, 37, 36, 59, 38, 57, 56, 39, 34, 61, 60, 35, 62, 33, 32, 63

Offset: 1

Yosu Yurramendi, Mar 27 2017

nonn

The inverse permutation is A284460.

Yosu Yurramendi, Table of n, a(n) for n = 1..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A002487, A020650, A020651, A162911, A162912, A245327, A245328, A284460.

maxrow <- 12 # by choice
a <- 1
b01 <- 1
for(m in 0:maxrow){
  b01 <- c(b01, c(1-b01[2^m:(2^(m+1)-1)], b01[2^m:(2^(m+1)-1)]) )
  for(k in 0:(2^m-1)){
    a[2^(m+1) +       k] <- a[2^m + k] + 2^(m + b01[2^(m+1) +       k])
    a[2^(m+1) + 2^m + k] <- a[2^m + k] + 2^(m + b01[2^(m+1) + 2^m + k])
}}
a
# Yosu Yurramendi, Mar 27 2017

maxblock <- 7 # by choice
a <- 1:3
for(n in 4:2^maxblock){
ones <- which(as.integer(intToBits(n)) == 1)
nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
anbit <- nbit
for(i in 2:(length(anbit) - 1))
   anbit[i] <- 1 - bitwXor(anbit[i], anbit[i-1])
a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
a
# Yosu Yurramendi, Apr 25 2021

a(n) = A258996(A231551(n)) = A231551(A092569(n)), n > 0 . - Yosu Yurramendi, Apr 10 2017

A317839 Möbius transform of A002487, Stern's Diatomic sequence.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 4, 0, 6, 0, 4, 0, 6, 0, 4, 0, 4, 0, 6, 0, 4, 0, 0, 0, 4, 0, 10, 0, 4, 0, 10, 0, 12, 0, 6, 0, 8, 0, 6, 0, 6, 0, 12, 0, 4, 0, 2, 0, 10, 0, 8, 0, -4, 0, 0, 0, 10, 0, 6, 0, 12, 0, 14, 0, 10, 0, 10, 0, 12, 0, 6, 0, 18, 0, 14, 0, 10, 0, 16, 0, 12, 0, 10, 0, 2, 0, 10, 0, 8, 0, 18, 0, 16, 0, 4

Offset: 1

Antti Karttunen, Aug 09 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Stern's sequences

Cf. A002487, A008683, A317837, A317838, A317840, A317841, A317843.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317839(n) = sumdiv(n,d,moebius(n/d)*A002487(d));

a(n) = Sum_{d|n} A008683(n/d)*A002487(d).

a(n) = A000010(n) - A317841(n).

A163659 L.g.f.: Sum_{n>=1} a(n)*x^n/n = log(S(x)/x) where S(x) is the g.f. of Stern's diatomic series (A002487).

Original entry on oeis.org

1, 3, -2, 7, 1, -6, 1, 15, -2, 3, 1, -14, 1, 3, -2, 31, 1, -6, 1, 7, -2, 3, 1, -30, 1, 3, -2, 7, 1, -6, 1, 63, -2, 3, 1, -14, 1, 3, -2, 15, 1, -6, 1, 7, -2, 3, 1, -62, 1, 3, -2, 7, 1, -6, 1, 15, -2, 3, 1, -14, 1, 3, -2, 127, 1, -6, 1, 7, -2, 3, 1, -30, 1, 3, -2, 7, 1, -6, 1, 31, -2, 3, 1, -14, 1

Offset: 1

Paul D. Hanna, Aug 02 2009

			L.g.f.: log(S(x)/x) = x + 3*x^2/2 - 2*x^3/3 + 7*x^4/4 + x^5/5 - 6*x^6/6 +...
where S(x) is the g.f. of Stern's diatomic series (A002487):
S(x) = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + 4*x^9 +...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A163658, A195587, A002487.

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 2, 1, # == 2, 2^(#2 + 1) - 1, # == 3, -2, True, 1] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 27 2017 *)

A002487(n)=local(c=1, b=0); while(n>0, if(bitand(n, 1), b+=c, c+=b); n>>=1); b
{a(n)=n*polcoeff(log(sum(k=0,n,A002487(k+1)*x^k)+x*O(x^n)),n)}
for(n=1, 64, print1(a(n), ", "))

{a(n) = if( n<1, 0, if( n%3, 1, -2) * sigma(2 ^ valuation(n, 2)))}; /* Michael Somos, Feb 14 2011 */

a(n)=local(X=x+x*O(x^n), A); A=log(1+X+X^2) + sum(k=0, #binary(n), log(1 + X^(2*2^k) + X^(4*2^k))); n*polcoeff(A, n)
for(n=1, 64, print1(a(n), ", ")) \\ Paul D. Hanna, May 04 2014

a(2^n) = 2^(n+1) - 1 for n>=0.
a(n) is multiplicative with a(2^e) = 2^(e+1) - 1, a(3^e) = -2 if e>0, a(p^e) = 1 if p>3. - Michael Somos, Feb 14 2011
L.g.f.: Sum_{n>=0} log(1 + x^(2^n) + x^(2*2^n)) = Sum_{n>=1} a(n)*x^n/n. - Paul D. Hanna, May 04 2014
G.f.: Sum_{n>=0} 2^n * x^(2^n) * (1 + 2*x^(2^n)) / (1 + x^(2^n) + x^(2*2^n)). - Paul D. Hanna, May 04 2014
Dirichlet g.f.: zeta(s) * (1 - 3^(1-s)) / (1 - 2^(1-s)). - Amiram Eldar, Oct 24 2023

A212289 Record values in A002487.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 15, 18, 19, 21, 23, 26, 29, 30, 34, 37, 41, 47, 49, 50, 55, 60, 67, 69, 76, 79, 89, 97, 108, 109, 123, 128, 129, 131, 144, 157, 175, 178, 181, 199, 207, 208, 233, 254, 257, 283, 287, 322, 335, 337, 338, 343, 377, 411, 458, 465, 467, 474

Offset: 1

Charles R Greathouse IV, Jun 13 2012

nonn

Distinct elements of A270362. - Jeffrey Shallit, Mar 16 2016.

Charles R Greathouse IV, Table of n, a(n) for n = 1..646
Ali Keramatipour and Jeffrey Shallit, Record-Setters in the Stern Sequence, arXiv:2205.06223 [math.CO], 2022.

Cf. A002487, A212288.

Union@ FoldList[Max, Nest[Append[#1, If[OddQ[#2], #1[[(#2 + 1)/2]], #1[[#2/2]] + #1[[(#2 + 2)/2]] ]] & @@ {#, Length@ # + 1} &, {0, 1}, 10^4]] (* Michael De Vlieger, Jul 10 2019 *)

fusc(n)=my(a=1, b=0); while(n, if(n%2, b+=a, a+=b); n>>=1); b
r=-1;for(n=0,1e5,t=fusc(n); if(t>r,r=t;print1(t", ")))

from itertools import count, islice
from functools import reduce
def A212289_gen(): # generator of terms
    yield (c:= 0)
    for n in count(1):
        m = sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0)))
        if m>c: yield (c:=m)
A212289_list = list(islice(A212289_gen(),30)) # Chai Wah Wu, Sep 23 2024

a(n) = fusc(A212288(n)).

a(1)=0 prepended in terms, b-file and programs by Georg Fischer, Jun 24 2020

A283986 a(n) = A002487(n-1) OR A002487(n), where OR is bitwise-or (A003986).

Original entry on oeis.org

1, 1, 3, 3, 3, 3, 3, 3, 5, 7, 7, 7, 7, 7, 7, 5, 5, 5, 7, 7, 11, 13, 7, 7, 7, 7, 13, 11, 7, 7, 5, 5, 7, 7, 13, 13, 15, 15, 15, 11, 11, 11, 13, 13, 13, 15, 15, 11, 11, 15, 15, 13, 13, 13, 11, 11, 11, 15, 15, 15, 13, 13, 7, 7, 7, 7, 15, 15, 15, 15, 13, 13, 15, 15, 27, 23, 23, 27, 15, 15, 15, 15, 27, 27, 29, 29, 31, 23, 21, 29, 31, 23, 23, 25, 11, 11, 11, 11, 25

Offset: 1

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Odd bisection of A283976.
Cf. A002487, A003986, A283988.
Cf. A283973 (positions where coincides with A007306, equally, with A283987).

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[BitOr[a[n - 1], a@ n], {n, 120}] (* 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)));
for(n=1, 101, print1(bitor(A(n - 1), A(n))", ")) \\ Indranil Ghosh, Mar 23 2017

from functools import reduce
def A283986(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0)))|sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n-1)[-1:2:-1],(1,0))) # Chai Wah Wu, May 05 2023

(define (A283986 n) (A003986bi (A002487 (- n 1)) (A002487 n))) ;; Where A003986bi implements bitwise-OR (A003986).

a(n) = A002487(n-1) OR A002487(n), where OR is bitwise-or (A003986).
a(n) = A283987(n) + A283988(n).
a(n) = A007306(n) - A283988(n).
a(n) = A283976((2*n)-1).

A283987 a(n) = A002487(n-1) XOR A002487(n), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

1, 0, 3, 3, 2, 1, 1, 2, 5, 7, 6, 7, 7, 6, 7, 5, 4, 1, 3, 4, 11, 13, 2, 5, 5, 2, 13, 11, 4, 3, 1, 4, 7, 3, 12, 13, 15, 12, 13, 9, 8, 3, 5, 8, 9, 11, 14, 11, 11, 14, 11, 9, 8, 5, 3, 8, 9, 13, 12, 15, 13, 12, 3, 7, 6, 1, 13, 14, 11, 7, 4, 9, 11, 4, 25, 21, 22, 27, 7, 14, 13, 5, 24, 27, 29, 24, 31, 23, 20, 29, 31, 20, 23, 25, 2, 9, 9, 2, 25, 23, 20, 31, 29, 20, 23

Offset: 1

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Odd bisection of A283977.
Cf. A002487, A003987, A283988.
Cf. A283973 (positions where coincides with A007306, or equally, with A283986).

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[BitXor[a[n - 1], a@ n], {n, 120}] (* 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)));
for(n=1, 120, print1(bitxor(A(n - 1), A(n)), ", ")) \\ Indranil Ghosh, Mar 23 2017

from functools import reduce
def A283987(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0)))^sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n-1)[-1:2:-1],(1,0))) if n>1 else 1 # Chai Wah Wu, May 05 2023

(define (A283987 n) (A003987bi (A002487 (- n 1)) (A002487 n))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(n) = A002487(n-1) XOR A002487(n), where XOR is bitwise-xor (A003987).
a(n) = A283986(n) - A283988(n).
a(n) = A007306(n) - 2*A283988(n).
a(n) = A283977((2*n)-1).

A283988 a(n) = A002487(n-1) AND A002487(n), where AND is bitwise-and (A004198).

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 4, 4, 3, 0, 0, 5, 2, 2, 5, 0, 0, 3, 4, 4, 1, 0, 4, 1, 0, 0, 3, 2, 2, 3, 8, 8, 5, 4, 4, 1, 0, 0, 1, 4, 4, 5, 8, 8, 3, 2, 2, 3, 0, 0, 1, 4, 0, 1, 6, 2, 1, 4, 8, 9, 4, 4, 11, 2, 2, 1, 0, 8, 1, 2, 10, 3, 0, 0, 5, 0, 0, 1, 0, 0, 3, 0, 0, 9, 2, 2, 9, 0, 0, 3, 0, 0, 1, 0, 0, 5, 0, 0, 3, 10, 2, 1, 8, 0, 1, 2, 2, 11, 4

Offset: 1

Antti Karttunen, Mar 21 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Odd bisection of A283978.
Cf. A002487, A004198, A007306, A283986, A283987.
Cf. A283973 (positions of zeros), A283974 (nonzeros).

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ@ n, a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[BitAnd[a[n - 1], a@ n], {n, 120}] (* 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)));
for(n=1, 120, print1(bitand(A(n - 1), A(n)),", ")) \\ Indranil Ghosh, Mar 23 2017

from functools import reduce
def A283988(n): return sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0)))&sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n-1)[-1:2:-1],(1,0))) if n>1 else 0 # Chai Wah Wu, May 05 2023

(define (A283988 n) (A004198bi (A002487 (- n 1)) (A002487 n)))  ;; Where A004198bi implements bitwise-AND (A004198).

a(n) = A002487(n-1) AND A002487(n), where AND is bitwise-and (A004198).
a(n) = A283986(n) - A283987(n).
a(n) = A007306(n) - A283986(n) = (A007306(n) - A283987(n))/2.
a(n) = A283978((2*n)-1).

A317837 a(n) = Sum_{d|n, dA002487(d).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 5, 1, 7, 1, 5, 6, 4, 1, 10, 1, 9, 6, 7, 1, 10, 4, 7, 7, 9, 1, 16, 1, 5, 8, 7, 7, 17, 1, 9, 8, 13, 1, 20, 1, 13, 14, 9, 1, 13, 4, 15, 8, 13, 1, 22, 9, 13, 10, 9, 1, 26, 1, 7, 18, 6, 9, 22, 1, 13, 10, 23, 1, 24, 1, 13, 17, 17, 9, 26, 1, 17, 15, 13, 1, 34, 9, 15, 10, 19, 1, 40, 9, 17, 8, 11, 11

Offset: 1

Antti Karttunen, Aug 09 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Stern's sequences

Cf. A002487, A293216, A317838, A317839, A317840, A317841, A317843.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317837(n) = sumdiv(n,d,(dA002487(d));

a(n) = Sum_{d|n, dA002487(d).
a(n) = A317838(n) - A002487(n).
a(n) = A001222(A293216(n)).

A317841 Möbius transform of A284013 (= n - A002487(n)), where A002487 is Stern's Diatomic sequence.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 4, 4, 4, 4, 6, 4, 8, 6, 8, 8, 12, 6, 12, 8, 8, 10, 16, 8, 16, 12, 14, 12, 22, 8, 26, 16, 20, 16, 20, 12, 26, 18, 20, 16, 30, 12, 30, 20, 18, 22, 38, 16, 36, 20, 26, 24, 40, 18, 36, 24, 34, 28, 48, 16, 52, 30, 40, 32, 48, 20, 56, 32, 38, 24, 58, 24, 58, 36, 30, 36, 50, 24, 66, 32, 48, 40, 64, 24, 50, 42

Offset: 1

Antti Karttunen, Aug 09 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Stern's sequences

Cf. A002487, A008683, A284013, A317839.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A284013(n) = (n - A002487(n));
A317841(n) = sumdiv(n,d,moebius(n/d)*A284013(d));

a(n) = Sum_{d|n} A008683(n/d)*A284013(d).

a(n) = A000010(n) - A317839(n).

A323889 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278222(i) = A278222(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 7, 5, 8, 2, 9, 6, 10, 4, 11, 7, 12, 3, 10, 7, 13, 5, 12, 8, 14, 2, 15, 9, 16, 6, 17, 10, 18, 4, 17, 11, 19, 7, 20, 12, 21, 3, 16, 10, 22, 7, 19, 13, 23, 5, 18, 12, 23, 8, 21, 14, 24, 2, 25, 15, 26, 9, 27, 16, 28, 6, 29, 17, 30, 10, 31, 18, 32, 4, 27, 17, 33, 11, 34, 19, 35, 7, 31, 20, 36, 12, 37, 21, 38, 3, 26, 16, 39, 10, 33, 22

Offset: 0

Antti Karttunen, Feb 09 2019

Restricted growth sequence transform of the ordered pair [A002487(n), A278222(n)].

Cf. A002487, A278222, A286622, A324400.

Cf. also A103391, A278243, A286378, A318311, A323892, A323897 and A324533 for a "deformed variant".

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux323889(n) = [A002487(n), A278222(n)];
v323889 = rgs_transform(vector(1+up_to,n,Aux323889(n-1)));
A323889(n) = v323889[1+n];

a(2^n) = 2 for all n >= 0.

cp's OEIS Frontend

A284459 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A002487/A002487' (Calkin-Wilf) into the enumeration system A245327/A245328, and A162911/A162912 (Drib) into A020651/A020650 (Yu-Ting inverted).

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A317839 Möbius transform of A002487, Stern's Diatomic sequence.

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A163659 L.g.f.: Sum_{n>=1} a(n)*x^n/n = log(S(x)/x) where S(x) is the g.f. of Stern's diatomic series (A002487).

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A212289 Record values in A002487.

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A283986 a(n) = A002487(n-1) OR A002487(n), where OR is bitwise-or (A003986).

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A283987 a(n) = A002487(n-1) XOR A002487(n), where XOR is bitwise-xor (A003987).

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A283988 a(n) = A002487(n-1) AND A002487(n), where AND is bitwise-and (A004198).

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