A002487 -id:A002487 - cp's OEIS frontend

A317840 Difference between Stern's Diatomic sequence (A002487) and its Möbius transform (A317839).

0, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 3, 4, 1, 1, 4, 1, 3, 4, 5, 1, 2, 3, 5, 4, 3, 1, 4, 1, 1, 6, 5, 5, 4, 1, 7, 6, 3, 1, 8, 1, 5, 6, 7, 1, 2, 3, 7, 6, 5, 1, 8, 7, 3, 8, 7, 1, 4, 1, 5, 10, 1, 7, 6, 1, 5, 8, 9, 1, 4, 1, 11, 8, 7, 7, 10, 1, 3, 8, 11, 1, 8, 7, 13, 8, 5, 1, 12, 7, 7, 6, 9, 9, 2, 1, 9, 8, 7, 1, 12, 1, 5, 14

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, A317837, A317838, A317839, A317841.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317840(n) = -sumdiv(n,d,(dA002487(d));

a(n) = -Sum_{d|n, dA008683(n/d)*A002487(d).

a(n) = A002487(n) - A317839(n).

A318306 Additive with a(p^e) = A002487(e).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 29 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. A002487, A077761, A318307.

Cf. also A046644.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318306(n) = vecsum(apply(e -> A002487(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318306(n): return sum(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(e)[-1:2:-1],(1,0))) for e in factorint(n).values()) # Chai Wah Wu, May 18 2023

a(n) = A007814(A318307(n)).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.15790080909728804399..., where f(x) = -x + x * (1-x) * Product{k>=0} (1 + x^(2^k) + x^(2^(k + 1))). - Amiram Eldar, Feb 11 2024

A324293 a(n) = A002487(sigma(n)).

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 5, 4, 2, 3, 3, 2, 2, 5, 4, 10, 3, 8, 1, 4, 2, 4, 5, 8, 3, 3, 4, 4, 1, 6, 2, 8, 2, 19, 7, 4, 3, 12, 8, 2, 5, 8, 10, 4, 2, 5, 10, 16, 4, 9, 8, 4, 4, 4, 3, 12, 4, 8, 5, 2, 5, 7, 8, 4, 5, 6, 2, 4, 4, 20, 11, 10, 5, 9, 2, 8, 3, 16, 13, 6, 8, 3, 8, 6, 4, 12, 12, 18, 3, 8, 1, 4, 4, 6, 9, 34, 10, 27, 12, 8, 5, 18, 2

Offset: 1

Antti Karttunen, Feb 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)
Index entries for sequences related to Stern's sequences

Cf. A000203, A002487, A324287, A324294.

A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ Modified from the one given in A002487, sign not actually needed here.
A324293(n) = A002487(sigma(n));

a(n) = A002487(A000203(n)).

A324337 a(n) = A002487(A006068(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 3, 4, 3, 2, 5, 1, 4, 5, 3, 5, 4, 3, 7, 2, 7, 8, 5, 1, 5, 7, 4, 7, 5, 3, 8, 6, 5, 4, 9, 3, 10, 11, 7, 2, 9, 12, 7, 11, 8, 5, 13, 1, 6, 9, 5, 10, 7, 4, 11, 9, 7, 5, 12, 3, 11, 13, 8, 7, 6, 5, 11, 4, 13, 14, 9, 3, 13, 17, 10, 15, 11, 7, 18, 2, 11, 16, 9, 17, 12, 7, 19, 14, 11, 8, 19, 5, 18, 21, 13, 1, 7, 11, 6, 13, 9, 5, 14, 13, 10

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Like in A324338, a few terms preceding each position n = 2^k seem to be a batch of nearby Fibonacci numbers in some order.

For all n > 0 A324338(n)/A324337(n) constitutes an enumeration system of all positive rationals. For all n > 0 A324338(n) + A324337(n) = A071585(n). - Yosu Yurramendi, Oct 22 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A002487, A006068, A063946, A071585, A248663, A283477, A324287, A324338.

Block[{f}, f[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b = a + b, a = a + b]; n = Floor[n/2]]; b]; Array[f@ Fold[BitXor, #, Quotient[#, 2^Range[BitLength[#] - 1]]] &, 106, 0]] (* Michael De Vlieger, Dec 14 2019, after Jean-François Alcover at A002487 and Jan Mangaldan at A006068 *)

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A324337(n) = A002487(A006068(n));

maxlevel <- 6 # by choice
#
b <- 0; A324338 <- 1; A324337 <- 1
for(i in 1:2^maxlevel) {
  b[2*i  ] <-     b[i]
  b[2*i+1] <- 1 - b[i]
  A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
  A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
  A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
  A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
#
A324338[1:127]; A324337[1:127]
# Yosu Yurramendi, Oct 22 2019

From Yosu Yurramendi, Oct 22 2019: (Start)
a(2^m+        k) = A324338(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1)
a(2^m+2^(m-1)+k) = A324338(2^m+        k), m > 0, 0 <= k < 2^(m-1). (End)
a(n) = A324338(A063946(n)), n > 0. Yosu Yurramendi, Nov 04 2019
a(n) = A002487(A248663(A283477(n))). - Antti Karttunen, Nov 06 2019
a(n) = A002487(1+A233279(n)). - Yosu Yurramendi, Nov 08 2019
From Yosu Yurramendi, Nov 28 2019: (Start)
a(2^(m+1)+k) - a(2^m+k) = A324338(k), m >= 0, 0 <= k < 2^m.
a(A059893(2^(m+1)+A001969(k+1))) - a(A059893(2^m+A001969(k+1))) = A071585(k), m >= 0, 0 <= k < 2^(m-1).
a(A059893(2^(m+1)+ A000069(k+1))) = A071585(k), m >= 1, 0 <= k < 2^(m-1). (End)
From Yosu Yurramendi, Nov 29 2019: (Start)
For n > 0:
A324338(n) + A324337(n) = A071585(n).
A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
a(n) = A002487(1+A233279(n)). - Yosu Yurramendi, Dec 27 2019

A324338 a(n) = A002487(1+A006068(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 2, 1, 4, 5, 3, 4, 3, 2, 5, 1, 5, 7, 4, 7, 5, 3, 8, 5, 4, 3, 7, 2, 7, 8, 5, 1, 6, 9, 5, 10, 7, 4, 11, 9, 7, 5, 12, 3, 11, 13, 8, 6, 5, 4, 9, 3, 10, 11, 7, 2, 9, 12, 7, 11, 8, 5, 13, 1, 7, 11, 6, 13, 9, 5, 14, 13, 10, 7, 17, 4, 15, 18, 11, 11, 9, 7, 16, 5, 17, 19, 12, 3, 14, 19, 11, 18, 13, 8, 21, 7, 6, 5, 11, 4, 13, 14, 9, 3, 13

Offset: 0

Antti Karttunen, Feb 23 2019

nonn

Like in A324337, a few terms preceding each 2^k-th term (here always 1) seem to consist of a batch of nearby Fibonacci numbers (A000045) in some order. For example, a(65533) = 987, a(65534) = 610 and a(65535) = 1597.

For all n > 0 A324338(n)/A324337(n) constitutes an enumeration system of all positive rationals. For all n > 0 A324338(n) + A324337(n) = A071585(n). - Yosu Yurramendi, Oct 22 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to Stern's sequences

Cf. A000045, A002487, A006068, A324288, A324337.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
A324338(n) = A002487(1+A006068(n));

maxlevel <- 6 # by choice #
b <- 0; A324338 <- 1; A324337 <- 1
for(i in 1:2^maxlevel) {
  b[2*i  ] <-     b[i]
  b[2*i+1] <- 1 - b[i]
  A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
  A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
  A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
  A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
#
A324338[1:127]; A324337[1:127]
# Yosu Yurramendi, Oct 22 2019

a(n) = A002487(1+A006068(n)).
a(2^n) = 1 for all n >= 0.
From Yosu Yurramendi, Oct 22 2019: (Start)
a(2^m+2^(m-1)+k) = A324337(2^m+        k), m > 0, 0 <= k < 2^(m-1)
a(2^m+        k) = A324337(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1). (End)
a(n) = A324337(A063946(n)), n > 0. Yosu Yurramendi, Nov 04 2019
a(n) = A002487(A233279(n)), n > 0. Yosu Yurramendi, Nov 08 2019
From Yosu Yurramendi, Nov 28 2019: (Start)
a(2^(m+1)+k) - a(2^m+k) = A324337(k), m >= 0,  0 <= k < 2^m.
a(A059893(2^(m+1)+A000069(k+1))) - a(A059893(2^m+A000069(k+1))) =  A071585(k), m >= 1,  0 <= k < 2^(m-1).
a(A059893(2^m+ A001969(k+1))) = A071585(k),    m >= 0,  0 <= k < 2^(m-1). (End)
From Yosu Yurramendi, Nov 29 2019: (Start)
For n > 0:
A324338(n) + A324337(n) = A071585(n).
A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
a(n) = A002487(A233279(n)). Yosu Yurramendi, Dec 27 2019

A070990 First differences of A002487.

Original entry on oeis.org

1, -1, 2, -1, 1, -2, 3, -1, 2, -3, 3, -2, 1, -3, 4, -1, 3, -4, 5, -3, 2, -5, 5, -2, 3, -5, 4, -3, 1, -4, 5, -1, 4, -5, 7, -4, 3, -7, 8, -3, 5, -8, 7, -5, 2, -7, 7, -2, 5, -7, 8, -5, 3, -8, 7, -3, 4, -7, 5, -4, 1, -5, 6, -1, 5, -6, 9, -5, 4, -9, 11, -4, 7, -11, 10, -7, 3, -10, 11, -3, 8, -11, 13, -8, 5, -13, 12, -5, 7, -12, 9, -7, 2, -9, 9, -2, 7, -9, 12, -7

Offset: 0

N. J. A. Sloane, May 24 2002

sign

Cf. A002487.

-a(2n+3) = a(2n) = A002487(n+2), n >= 0. - Ralf Stephan, Aug 17 2003

G.f.: -1/x^2 + ((1 - x)/x^2)*Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Feb 28 2017

A071884 Trajectory of 37 under map x -> A002487(x)*A002487(x+1).

Original entry on oeis.org

37, 77, 170, 714, 3450, 10414, 68145, 303610, 721305, 1815066, 17782782, 4598150, 68178460, 133118649, 585506070, 20663527908, 777507763200, 264506067126, 73112062114130, 5907184479605316, 516068243814152148

Offset: 0

N. J. A. Sloane and J. H. Conway, Jun 10 2002

nonn

All smaller starting values lead into a cycle after a small number of steps; this appears to have an infinite trajectory.

Cf. A002487, A071885, A071886, A071887.

# n is starting value, m is length of trajectory to compute
f := proc(n,m) local t1,i,j; t1 := [n]; for i from 2 to m do j := t1[i-1]; t1 := [op(t1), A002487(j)*A002487(j+1)]; od: t1; end;

A071885 Trajectory of 38 under map x -> A002487(x)*A002487(x+1).

Original entry on oeis.org

38, 70, 117, 198, 368, 210, 558, 1312, 693, 3920, 2438, 6900, 17138, 43792, 105939, 553150, 862695, 5466578, 262720694, 8088309190, 155799611901, 2918284055518, 415202250061643, 1166813484199476, 224311803609338424

Offset: 0

N. J. A. Sloane and J. H. Conway, Jun 10 2002

nonn

Cf. A002487, A071884, A071886, A071887.

A071886 Trajectory of 41 under map x -> A002487(x)*A002487(x+1).

Original entry on oeis.org

41, 88, 85, 273, 456, 370, 897, 418, 943, 561, 1240, 2457, 13860, 47652, 208380, 234724, 1687203, 8509066, 67089945, 13299944, 72454532, 940541150, 820208766, 5192523560, 13313765208, 46414871454, 6589784048694, 93614939731600

Offset: 0

N. J. A. Sloane and J. H. Conway, Jun 10 2002

nonn

Cf. A002487, A071885, A071884, A071887.

A071887 Conjectured values of n whose trajectory under map x -> A002487(x)*A002487(x+1) does not go into a cycle.

Original entry on oeis.org

37, 38, 41, 54, 57, 58

Offset: 1

J. H. Conway, Jun 10 2002

Why isn't 42 in this list? It does not enter a cycle in under 100 iterations which is as far as I could check. [Sean A. Irvine, Oct 05 2011]

Cf. A002487, A070871, A071884, A071885, A071886.

cp's OEIS Frontend

A317840 Difference between Stern's Diatomic sequence (A002487) and its Möbius transform (A317839).

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A318306 Additive with a(p^e) = A002487(e).

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A324293 a(n) = A002487(sigma(n)).

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A324337 a(n) = A002487(A006068(n)).

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Mathematica

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R

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A324338 a(n) = A002487(1+A006068(n)).

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A070990 First differences of A002487.

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A071884 Trajectory of 37 under map x -> A002487(x)*A002487(x+1).

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Maple

A071885 Trajectory of 38 under map x -> A002487(x)*A002487(x+1).

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A071886 Trajectory of 41 under map x -> A002487(x)*A002487(x+1).

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A071887 Conjectured values of n whose trajectory under map x -> A002487(x)*A002487(x+1) does not go into a cycle.

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