A004718 -id:A004718 - cp's OEIS frontend

A323887 Sum of Per Nørgård's "infinity sequence" (A004718) and its Dirichlet inverse (A323886).

2, 0, 0, 1, 0, -4, 0, -1, 4, 0, 0, 2, 0, -6, 0, 1, 0, 0, 0, 0, 12, -2, 0, -2, 0, 2, 0, 3, 0, -8, 0, -1, 4, 0, 0, 2, 0, -6, -4, 0, 0, 10, 0, 1, 16, -4, 0, 2, 9, -6, 0, -1, 0, 0, 0, -3, 12, 4, 0, 4, 0, -10, -20, 1, 0, 0, 0, 0, 8, -2, 0, -2, 0, 2, 12, 3, 6, -12, 0, 0, -4, -2, 0, 1, 0, -4, -8, -1, 0, 16, -6, 2, 20, -6, 0, -2, 0, 11, 0, 3, 0, -8, 0, 1, 28

Offset: 1

Antti Karttunen, Feb 08 2019

sign

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A004718, A323886.

Cf. also A323365, A323882, A323885, A323896.

up_to = 65537;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA004718list(up_to);
A004718(n) = v004718[n];
v323886 = DirInverse(v004718);
A323886(n) = v323886[n];
A323887(n) = (A004718(n)+A323886(n));

a(n) = A004718(n) + A323886(n).

A083866 Positions of zeros in Per Nørgård's infinity sequence (A004718).

Original entry on oeis.org

0, 5, 10, 17, 20, 27, 34, 40, 45, 54, 65, 68, 75, 80, 85, 90, 99, 105, 108, 119, 130, 136, 141, 150, 160, 165, 170, 177, 180, 187, 198, 210, 216, 221, 238, 257, 260, 267, 272, 277, 282, 291, 297, 300, 311, 320, 325, 330, 337, 340, 347, 354, 360

Offset: 0

Ralf Stephan, May 07 2003

nonn

First differences seem to be always >2.
Many (but not all) prime members are in A005107.
The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Yu Hin (Gary) Au, Christopher Drexler-Lemire, and Jeffrey Shallit, Notes and note pairs in Norgard's infinity series, Journal of Mathematics and Music, Volume 11, Issue 1, 2017, pages 1-19.

Cf. A004718, A005107.

a083866 n = a083866_list !! n
a083866_list = filter ((== 0) . a004718) [0..]
-- Reinhard Zumkeller, Mar 19 2015, Nov 10 2012

from itertools import groupby, islice
def A083866_gen(startvalue=0): # generator of terms >= startvalue
    n, c = max(0,startvalue),0
    for k, g in groupby(bin(n)[2:]):
        c = c+len(list(g)) if k == '1' else (-c if len(list(g))&1 else c)
    while True:
        if c == 0: yield n
        n += 1
        c = c-t-1 if (t:=(~n & n-1).bit_length())&1 else t+1-c
A083866_list = list(islice(A083866_gen(),20)) # Chai Wah Wu, Mar 02 2023

A323909 Balanced ternary representation of A004718, Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 1, 2, 5, 1, 0, 7, 3, 2, 5, 0, 1, 5, 2, 6, 4, 1, 0, 7, 3, 0, 1, 2, 5, 7, 3, 1, 0, 3, 7, 8, 17, 2, 5, 0, 1, 5, 2, 6, 4, 0, 1, 2, 5, 1, 0, 7, 3, 5, 2, 6, 4, 2, 5, 0, 1, 6, 4, 5, 2, 4, 6, 22, 15, 1, 0, 7, 3, 0, 1, 2, 5, 7, 3, 1, 0, 3, 7, 8, 17, 0, 1, 2, 5, 1, 0, 7, 3, 2, 5, 0, 1, 5, 2, 6, 4, 7, 3, 1, 0, 3, 7, 8, 17, 1, 0

Offset: 0

Antti Karttunen, Feb 10 2019

nonn

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Antti Karttunen, Table of n, a(n) for n = 0..65536

Cf. A004718, A083866 (positions of zeros), A117966, A117967, A117968, A323907 (rgs-transform), A323908.

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A323909(n) = { my(x = A004718(n)); if(x >= 0,A117967(x),A117968(-x)); };

If A004718(n) >= 0, then a(n) = A117967(A004718(n)), otherwise a(n) = A117968(-A004718(n)).

For all n >= 1, A117966(a(n)) = A004718(n).

A323886 Dirichlet inverse of A004718, Per Nørgård's "infinity sequence".

Original entry on oeis.org

1, 1, -2, 0, 0, -2, -3, 0, 2, 0, -1, 0, 1, -3, -4, 0, 0, 2, -3, 0, 11, -1, -2, 0, -3, 1, 0, 0, 2, -4, -5, 0, 2, 0, -1, 0, 1, -3, -8, 0, -1, 11, -2, 0, 16, -2, -3, 0, 10, -3, -4, 0, -2, 0, -1, 0, 8, 2, 1, 0, 3, -5, -26, 0, 0, 2, -3, 0, 7, -1, -2, 0, -3, 1, 12, 0, 8, -8, -5, 0, -5, -1, -2, 0, 0, -2, -11, 0, -2, 16, -7, 0, 21, -3, -4, 0, -3, 10, 0, 0, 2, -4, -5, 0

Offset: 1

Antti Karttunen, Feb 08 2019

sign

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A004718, A323881, A323887.

b[0] = 0;
b[n_?EvenQ] := b[n] = -b[n/2];
b[n_] := b[n] = b[(n - 1)/2] + 1;
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 16 2020 *)

up_to = 65537;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, v[n>>1]+1, -v[n/2])); (v); }; \\ After code in A004718.
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA004718list(up_to));
A323886(n) = v323886[n];

A255723 Another variant of Per Nørgård's "infinity sequence", cf. A004718: t(0) = 0; t(4n) = t(n); t(4n+1) = t(n) - 2; t(4n+2) = -t(n) - 1; t(4n+3) = t(n) + 2.

Original entry on oeis.org

0, -2, -1, 2, -2, -4, 1, 0, -1, -3, 0, 1, 2, 0, -3, 4, -2, -4, 1, 0, -4, -6, 3, -2, 1, -1, -2, 3, 0, -2, -1, 2, -1, -3, 0, 1, -3, -5, 2, -1, 0, -2, -1, 2, 1, -1, -2, 3, 2, 0, -3, 4, 0, -2, -1, 2, -3, -5, 2, -1, 4, 2, -5, 6, -2, -4, 1, 0, -4, -6, 3, -2, 1, -1

Offset: 0

Reinhard Zumkeller, Mar 19 2015

sign

Per Nørgård's surname is also written as Noergaard;

example of a sequence sharing with A004718 some main characterizing properties, see link (chapter 7).

Yu Hin (Gary) Au, Christopher Drexler-Lemire, and Jeffrey Shallit, "Notes and note pairs in Norgard's infinity series", J. of Mathematics and Music (2017). DOI: http://dx.doi.org/10.1080/17459737.2017.1299807

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Christopher Drexler-Lemire, Jeffrey Shallit, Notes and Note-Pairs in Noergaard's Infinity Series, arXiv:1402.3091 [math.CO], page 12f.
Jørgen Mortensen, "Is Per Nørgård's infinity series unique?"

Cf. A004718, A256184, A087808, A085144.

a255723 n = a255723_list !! n
a255723_list = 0 : concat (transpose [map (subtract 2) a255723_list,
                                      map (-1 -) a255723_list,
                                      map (+ 2) a255723_list,
                                      tail a255723_list])

A256184 First of two variations by Per Nørgård of his "infinity sequence", cf. A004718: u(0) = 0; u(3n) = -u(n); u(3n+1) = u(n) - 2; u(3*n+2) = u(n) - 1.

Original entry on oeis.org

0, -2, -1, 2, -4, -3, 1, -3, -2, -2, 0, 1, 4, -6, -5, 3, -5, -4, -1, -1, 0, 3, -5, -4, 2, -4, -3, 2, -4, -3, 0, -2, -1, -1, -1, 0, -4, 2, 3, 6, -8, -7, 5, -7, -6, -3, 1, 2, 5, -7, -6, 4, -6, -5, 1, -3, -2, 1, -3, -2, 0, -2, -1, -3, 1, 2, 5, -7, -6, 4, -6, -5

Offset: 0

Reinhard Zumkeller, Mar 19 2015

sign

Per Nørgård's surname is also written as Noergaard.

Not squarefree in contrast to A004718, first repetition of order 3: a(32) = a(33) = a(34) = -1, see link.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Yu Hin (Gary) Au, Christopher Drexler-Lemire, and Jeffrey Shallit, Notes and note pairs in Norgard's infinity series, J. of Mathematics and Music (2017).
Christopher Drexler-Lemire and Jeffrey Shallit, Notes and Note-Pairs in Noergaard's Infinity Series, arXiv:1402.3091 [math.CO], 2014, page 13.

Cf. A004718, A256185, A087808, A085144, A255723.

a256184 n = a256184_list !! n
a256184_list = 0 : concat (transpose [map (subtract 2) a256184_list,
                                      map (subtract 1) a256184_list,
                                      map negate $ tail a256184_list])

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n): return 0 if n == 0 else (a(n//3) - (3-n%3)) if n%3 else -a(n//3)
print([a(n) for n in range(72)]) # Michael S. Branicky, Sep 02 2021

A323907 Lexicographically earliest positive sequence such that a(i) = a(j) => A004718(i) = A004718(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 09 2019

nonn

Restricted growth sequence transform of A004718, Per Nørgård's "infinity sequence".

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Antti Karttunen, Table of n, a(n) for n = 0..65535

Restricted growth sequence transform of A004718, A323908 and A323909.

Cf. A083866 (positions of ones).

up_to = 65535;
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; };
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
v323907 = rgs_transform(vector(1+up_to,n,A004718(n-1)));
A323907(n) = v323907[1+n];

A323908 Reversing binary representation of A004718, Per Nørgård's "infinity sequence".

Original entry on oeis.org

0, 1, 3, 2, 1, 0, 6, 7, 3, 2, 0, 1, 2, 3, 5, 4, 1, 0, 6, 7, 0, 1, 3, 2, 6, 7, 1, 0, 7, 6, 12, 13, 3, 2, 0, 1, 2, 3, 5, 4, 0, 1, 3, 2, 1, 0, 6, 7, 2, 3, 5, 4, 3, 2, 0, 1, 5, 4, 2, 3, 4, 5, 15, 14, 1, 0, 6, 7, 0, 1, 3, 2, 6, 7, 1, 0, 7, 6, 12, 13, 0, 1, 3, 2, 1, 0, 6, 7, 3, 2, 0, 1, 2, 3, 5, 4, 6, 7, 1, 0, 7, 6, 12, 13, 1, 0

Offset: 0

Antti Karttunen, Feb 09 2019

nonn

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A004718, A048724, A065620, A065621, A083866 (positions of zeros), A323907 (rgs-transform), A323909.

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A048724(n) = bitxor(n, n<<1);
A065621(n) = bitxor(n-1,n+n-1);
A323908(n) = if(A004718(n)<=0, A048724(-A004718(n)), A065621(A004718(n)));

from itertools import groupby
def A323908(n):
    c = 0
    for k, g in groupby(bin(n)[2:]):
        c = c+len(list(g)) if k == '1' else (-c if len(list(g))&1 else c)
    return -c^(-c<<1) if c<=0 else c^(c&~-c)<<1 # Chai Wah Wu, Mar 02 2023

If A004718(n) <= 0, a(n) = A048724(-A004718(n)), otherwise a(n) = A065621(A004718(n)).

For all n >= 1, A065620(a(n)) = A004718(n).

A256185 Second of two variations by Per Nørgård of his "infinity sequence", cf. A004718: v(0) = 0; v(3n) = -v(n); v(3n+1) = v(n) - 3; v(3*n+2) = -2 - v(n).

Original entry on oeis.org

0, -3, -2, 3, -6, 1, 2, -5, 0, -3, 0, -5, 6, -9, 4, -1, -2, -3, -2, -1, -4, 5, -8, 3, 0, -3, -2, 3, -6, 1, 0, -3, -2, 5, -8, 3, -6, 3, -8, 9, -12, 7, -4, 1, -6, 1, -4, -1, 2, -5, 0, 3, -6, 1, 2, -5, 0, 1, -4, -1, 4, -7, 2, -5, 2, -7, 8, -11, 6, -3, 0, -5, 0

Offset: 0

Reinhard Zumkeller, Mar 19 2015

sign

Per Nørgård's surname is also written as Noergaard;

for all odd j exists k such that abs(a(k+1)-a(k)) = j, in contrast to A004718, where this holds also for even j > 0, see link.

Yu Hin (Gary) Au, Christopher Drexler-Lemire, and Jeffrey Shallit, "Notes and note pairs in Norgard's infinity series", J. of Mathematics and Music (2017). DOI: http://dx.doi.org/10.1080/17459737.2017.1299807

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Christopher Drexler-Lemire, Jeffrey Shallit, Notes and Note-Pairs in Noergaard's Infinity Series, arXiv:1402.3091 [math.CO], page 13

Cf. A004718, A256184, A087808, A085144, A255723.

a256185 n = a256185_list !! n
a256185_list = 0 : concat (transpose [map (subtract 3) a256185_list,
                                      map (-2 -) a256185_list,
                                      map negate $ tail a256185_list])

A325803 Nonzero terms of Product_{k=0..floor(log_2(n))} (1 + A004718(floor(n/(2^k)))).

Original entry on oeis.org

1, 2, 6, -6, 24, -18, -48, 120, 18, -72, -192, 48, -360, 720, 54, 144, -360, 384, -960, 144, -1800, 720, -2880, 5040, -54, 216, 576, -144, 1080, -2160, 1536, -384, 2880, -5760, -144, 576, 5400, -10800, 2880, -720, -17280, 8640, -25200, 40320, -162, -432, 1080

Offset: 1

Mikhail Kurkov, May 22 2019

See A329893.

Mikhail Kurkov, Table of n, a(n) for n = 1..13495

Cf. A000120, A001405, A004718, A284005, A325804, A329893.

a[n_?EvenQ] := a[n] = -a[n/2]; a[0] = 0; a[n_] := a[n] = a[(n - 1)/2] + 1; DeleteCases[Table[Product[ 1 + a[Floor[n/(2^k)]], {k, 0, Floor[Log2[n]]}], {n, 0, 200}], 0] (* Michael De Vlieger, Apr 22 2024, after Jean-François Alcover at A004718 *)

b(n) = if(n==0, 0, (-1)^(n+1)*b(n\2) + n%2); \\ A004718
f(n) = if(n==0, 1, prod(k=0, logint(n,2), 1+b(n\2^k)));
lista(nn) = for (n=0, nn, if (f(n), print1(f(n), ", "))); \\ Michel Marcus, May 26 2019

from itertools import count, islice
from math import prod
def A325803_gen(): # generator of terms
    for n in count(0):
        c, s = [0]*(m:=n.bit_length()), bin(n)[2:]
        for i in range(m):
            if s[i]=='1':
                for j in range(m-i):
                    c[j] = c[j]+1
            else:
                for j in range(m-i):
                    c[j] = -c[j]
        if (k:=prod(1+d for d in c)): yield k
A325803_list = list(islice(A325803_gen(),20)) # Chai Wah Wu, Mar 03 2023

a(n) = A329893(A325804(n)). - Antti Karttunen, Dec 10 2019

Comments and two formulas moved to A329893, which is an "uncompressed" version of this sequence. - Antti Karttunen, Dec 11 2019

cp's OEIS Frontend

A323887 Sum of Per Nørgård's "infinity sequence" (A004718) and its Dirichlet inverse (A323886).

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A083866 Positions of zeros in Per Nørgård's infinity sequence (A004718).

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A323909 Balanced ternary representation of A004718, Per Nørgård's "infinity sequence".

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A323886 Dirichlet inverse of A004718, Per Nørgård's "infinity sequence".

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A255723 Another variant of Per Nørgård's "infinity sequence", cf. A004718: t(0) = 0; t(4*n) = t(n); t(4*n+1) = t(n) - 2; t(4*n+2) = -t(n) - 1; t(4*n+3) = t(n) + 2.

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A256184 First of two variations by Per Nørgård of his "infinity sequence", cf. A004718: u(0) = 0; u(3*n) = -u(n); u(3*n+1) = u(n) - 2; u(3*n+2) = u(n) - 1.

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A323907 Lexicographically earliest positive sequence such that a(i) = a(j) => A004718(i) = A004718(j), for all i, j >= 0.

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A323908 Reversing binary representation of A004718, Per Nørgård's "infinity sequence".

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A255723 Another variant of Per Nørgård's "infinity sequence", cf. A004718: t(0) = 0; t(4n) = t(n); t(4n+1) = t(n) - 2; t(4n+2) = -t(n) - 1; t(4n+3) = t(n) + 2.

A256184 First of two variations by Per Nørgård of his "infinity sequence", cf. A004718: u(0) = 0; u(3n) = -u(n); u(3n+1) = u(n) - 2; u(3*n+2) = u(n) - 1.

A256185 Second of two variations by Per Nørgård of his "infinity sequence", cf. A004718: v(0) = 0; v(3n) = -v(n); v(3n+1) = v(n) - 3; v(3*n+2) = -2 - v(n).