A002487 -id:A002487 - cp's OEIS frontend

A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.

1, 1, 1, 3, 3, 1, 3, 5, 3, 3, 5, 3, 5, 3, 1, 35, 5, 3, 7, 9, 5, 5, 7, 5, 19, 5, 5, 9, 7, 1, 5, 63, 1, 5, 9, 9, 11, 7, 5, 15, 11, 5, 13, 15, 13, 7, 9, 35, 27, 19, 7, 15, 13, 5, 7, 15, 3, 7, 11, 3, 9, 5, -7, 231, -1, 1, 11, 15, 7, 9, 13, 15, 15, 11, 47, 21, 19, 5, 13, 105, 27, 11, 19, 15, 27, 13, 11, 25, 17, 13, 23, 21, 11, 9, 1, 63

Offset: 1

Antti Karttunen, Aug 11 2018

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

Cf. A002487, A317932 (denominators, conjectured).

Cf. also A299151, A317927, A317839, A317843.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317931perA317932(n) = if(1==n,n,(A002487(n)-sumdiv(n,d,if((d>1)&&(dA317931perA317932(d)*A317931perA317932(n/d),0)))/2);
A317931(n) = numerator(A317931perA317932(n));

\\ Memoized implementation:
memo = Map();
A317931perA317932(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (A002487(n)-sumdiv(n,d,if((d>1)&&(dA317931perA317932(d)*A317931perA317932(n/d),0)))/2); mapput(memo,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A002487(n) - Sum_{d|n, d>1, d 1.

A318509 Completely multiplicative with a(p) = A002487(p).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 6, 1, 5, 4, 7, 3, 6, 5, 7, 2, 9, 5, 8, 3, 7, 6, 5, 1, 10, 5, 9, 4, 11, 7, 10, 3, 11, 6, 13, 5, 12, 7, 9, 2, 9, 9, 10, 5, 13, 8, 15, 3, 14, 7, 11, 6, 9, 5, 12, 1, 15, 10, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 15, 10, 13, 3, 16, 11, 19, 6, 15, 13, 14, 5, 17, 12, 15, 7, 10, 9, 21, 2, 11, 9, 20, 9, 19, 10, 17, 5, 18

Offset: 1

Antti Karttunen, Aug 30 2018

Provided that the conjecture given in A261179 holds, then for all n >= 1, A007814(a(n)) = A007949(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A261179, A318510.

Cf. also A318307.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318509(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(f[i, 1])); factorback(f); };

from math import prod
from functools import reduce
from sympy import factorint
def A318509(n): return prod(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(p)[-1:2:-1],(1,0)))**e for p, e in factorint(n).items()) # Chai Wah Wu, May 18 2023

A319687 a(n) = A318509(n) - A002487(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 4, 0, 4, 0, 0, 2, 0, 0, 6, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, -2, -6, 0, -4, 0, 0, 0, -4, 0, -6, 0, 10, 0, 0, 0, 4, 2, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 02 2018

All terms seem to be even. See the conjecture given in A261179.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A261179, A317837, A318509, A323365.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318509(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(f[i, 1])); factorback(f); };
A319687(n) = (A318509(n) - A002487(n));

from math import prod
from functools import reduce
from sympy import factorint
def A319687(n): return prod(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(p)[-1:2:-1],(1,0)))**e for p, e in factorint(n).items())-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))) # Chai Wah Wu, May 18 2023

a(n) = A318509(n) - A002487(n).

A323897 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A083254(i) = A083254(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 8, 12, 2, 13, 14, 15, 16, 17, 18, 19, 20, 21, 18, 22, 16, 23, 24, 25, 2, 26, 18, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 30, 38, 39, 40, 41, 42, 36, 43, 34, 44, 32, 45, 30, 46, 47, 48, 18, 49, 2, 50, 51, 52, 36, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 58, 66, 67, 68, 69, 70, 71, 72, 73, 74, 60, 75

Offset: 1

Antti Karttunen, Feb 09 2019

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A083254(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A000010, A002487, A083254.

Cf. also A323892, A323898.

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
A083254(n) = (2*eulerphi(n)-n);
A323897aux(n) = [A002487(n), A083254(n)];
v323897 = rgs_transform(vector(up_to,n,A323897aux(n)));
A323897(n) = v323897[n];

a(2^n) = 2 for all n >= 1.

A324049 a(n) = A002487(A324051(n)).

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 2, 0, 1, 4, 3, 2, 1, 1, 2, 11, 2, 1, 4, 4, 4, 7, 5, 1, 1, 1, 5, 3, 3, 26, 4, 3, 11, 4, 3, 2, 1, 2, 5, 21, 4, 3, 6, 1, 5, 37, 7, 2, 5, 1, 7, 1, 3, 12, 9, 4, 40, 14, 8, 4, 76, 1, 6, 4, 8, 63, 5, 7, 3, 3, 1, 53, 27, 3, 9, 7, 5, 5, 10, 2, 199, 98, 10, 4, 37, 6, 4, 2, 4, 8, 25, 34, 104, 12, 3, 43, 6, 1, 4, 5, 2, 117, 13, 2, 1, 3, 7, 76, 4

Offset: 2

Antti Karttunen, Feb 22 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 2..4473

Cf. A002487, A324051.

Cf. also A323902, A324115, A324116.

PARI
```
A324049(n) = A002487(A324051(n));
```

a(n) = A002487(A324051(n)).

A324286 a(n) = A002487(A048675(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 4, 2, 1, 1, 3, 1, 2, 3, 5, 1, 3, 1, 6, 2, 3, 1, 3, 1, 3, 4, 7, 2, 2, 1, 8, 5, 3, 1, 5, 1, 4, 1, 9, 1, 2, 1, 4, 6, 5, 1, 3, 3, 5, 7, 10, 1, 1, 1, 11, 2, 2, 4, 7, 1, 6, 8, 5, 1, 3, 1, 12, 3, 7, 2, 9, 1, 1, 1, 13, 1, 2, 5, 14, 9, 7, 1, 4, 3, 8, 10, 15, 6, 3, 1, 5, 3, 3, 1, 11, 1, 9, 3

Offset: 1

Antti Karttunen, Feb 22 2019

nonn

Like A323902 and A323903, this also has quite a moderate growth rate, even though some terms of A048675 soon grow quite big.

Cf. A002487, A048675, A283477, A322821, A323902, A323903, A324287.

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); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A324286(n) = A002487(A048675(n));

a(n) = A002487(A048675(n)) = A002487(A322821(n)).

a(A283477(n)) = A324287(n).

A331601 a(n) = A002487(A241909(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 3, 2, 8, 1, 7, 1, 14, 4, 3, 1, 4, 1, 11, 8, 22, 1, 9, 2, 64, 3, 43, 1, 18, 1, 5, 14, 110, 4, 9, 1, 162, 22, 47, 1, 34, 1, 127, 7, 440, 1, 13, 2, 12, 64, 191, 1, 8, 8, 97, 110, 1002, 1, 23, 1, 752, 11, 5, 14, 112, 1, 1249, 162, 16, 1, 17, 1, 610, 4, 897, 4, 220, 1, 111, 3, 4882, 1, 121, 22, 5494, 440, 281, 1, 26, 8, 7623, 1002

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

Cf. A002487, A241909, A331732.

Cf. also A323901, A323902, A323903, A331600.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A331601(n) = A002487(A241909(n));

a(n) = A002487(A241909(n)).

a(n) = A002487(A331732(n)).

A331748 a(n) = A002487(n) XOR A002487(A163511(n)).

Original entry on oeis.org

1, 0, 0, 0, 0, 7, 0, 0, 0, 12, 7, 2, 0, 1, 0, 7, 0, 11, 12, 12, 7, 26, 2, 14, 0, 11, 1, 1, 0, 15, 7, 0, 0, 16, 11, 34, 12, 41, 12, 37, 7, 27, 26, 26, 2, 22, 14, 4, 0, 25, 11, 27, 1, 31, 1, 26, 0, 12, 15, 0, 7, 15, 0, 3, 0, 71, 16, 116, 11, 126, 34, 108, 12, 91, 41, 107, 12, 11, 37, 98, 7, 76, 27, 74, 26, 61, 26, 43, 2, 53, 22, 42, 14, 34, 4, 22, 0, 57

Offset: 0

Antti Karttunen, Feb 05 2020

Cf. A002487, A003987, A163511, A323901, A331743, A331749.

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 }; \\ From A005940
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A331748(n) = bitxor(A002487(A163511(n)), A002487(n));

a(n) = A002487(n) XOR A323901(n) = A002487(n) XOR A002487(A163511(n)).

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

A331749 a(n) = A002487(A163511(n)) - A002487(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 4, 1, 2, 0, -1, 0, -1, 0, 9, 4, 4, 1, 10, 2, 2, 0, 5, -1, 1, 0, 1, -1, 0, 0, 16, 9, 34, 4, 23, 4, 37, 1, 5, 10, 10, 2, 14, 2, 4, 0, 7, 5, 11, -1, 5, 1, 6, 0, -4, 1, 0, -1, -3, 0, -1, 0, 57, 16, 116, 9, 98, 34, 84, 4, 69, 23, 103, 4, 9, 37, 98, 1, 52, 5, 70, 10, 19, 10, 39, 2, 19, 14, 38, 2, 34, 4, 18, 0, 39

Offset: 0

Antti Karttunen, Feb 05 2020

sign

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, A163511, A323901, A331743, A331748.

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 }; \\ From A005940
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A331749(n) = (A002487(A163511(n)) - A002487(n));

a(n) = A323901(n) - A002487(n) = A002487(A163511(n)) - A002487(n).

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

A355090 Square array A(n, k), n >= 0, k > 0, read by antidiagonals upwards; A(n, k) is the unique m such that n/k = fusc(m)/fusc(m+1) (where fusc is Stern's diatomic series A002487).

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 1, 4, 0, 15, 5, 6, 8, 0, 31, 3, 1, 2, 16, 0, 63, 11, 9, 14, 12, 32, 0, 127, 7, 13, 1, 10, 4, 64, 0, 255, 23, 3, 17, 30, 2, 24, 128, 0, 511, 15, 19, 5, 1, 6, 28, 8, 256, 0, 1023, 47, 27, 29, 33, 62, 18, 20, 48, 512, 0, 2047, 31, 7, 3, 25, 1, 22, 2, 4, 16, 1024, 0

Offset: 0

Rémy Sigrist, Jun 18 2022

The binary expansion of A(n, k) encodes the position of n/k (> 0) in the Calkin-Wilf tree.

			Square array A(n, k) begins:
  n\k|      1    2    3    4    5    6    7    8    9    10    11    12
  ---+-----------------------------------------------------------------
    0|      0    0    0    0    0    0    0    0    0     0     0     0
    1|      1    2    4    8   16   32   64  128  256   512  1024  2048
    2|      3    1    6    2   12    4   24    8   48    16    96    32
    3|      7    5    1   14   10    2   28   20    4    56    40     8
    4|     15    3    9    1   30    6   18    2   60    12    36     4
    5|     31   11   13   17    1   62   22   26   34     2   124    44
    6|     63    7    3    5   33    1  126   14    6    10    66     2
    7|    127   23   19   29   25   65    1  254   46    38    58    50
    8|    255   15   27    3   21    9  129    1  510    30    54     6
    9|    511   47    7   35   61    5   49  257    1  1022    94    14
   10|   1023   31   39   11    3   13   57   17  513     1  2046    62
   11|   2047   95   55   59   67  125   37   41   97  1025     1  4094
   12|   4095   63   15    7   51    3   45    5    9    33  2049     1

Rémy Sigrist, Table of n, a(n) for n = 0..10152
Wikipedia, Calkin-Wilf tree
Index entries for sequences related to Stern's sequences

Cf. A000918, A002487, A054429.

A(x,y) = { if (x==0, 0, my (v=0,t=1,a=0,b=1,c=1,d=0); while (1, my (m=a+c, n=b+d); if (x*n==y*m, return (t+v), x*n

A(m*n, m*k) = A(n, k) for any m > 0.
A(k, n) = A054429(A(n, k)) for any n, k > 0.
A(0, k) = 0.
A(1, k) = 2^(k-1).
A(n, 1) = 2^n - 1.
A(n, n+1) = A000918(n+1).
A(A002487(n), A002487(n+1)) = n.

cp's OEIS Frontend

A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.

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A318509 Completely multiplicative with a(p) = A002487(p).

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A319687 a(n) = A318509(n) - A002487(n).

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A323897 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A083254(i) = A083254(j), for all i, j >= 1.

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A324049 a(n) = A002487(A324051(n)).

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A324286 a(n) = A002487(A048675(n)).

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A331601 a(n) = A002487(A241909(n)).

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A331748 a(n) = A002487(n) XOR A002487(A163511(n)).

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A331749 a(n) = A002487(A163511(n)) - A002487(n).