A317932 -id:A317932 - 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.

A318666 a(n) = 2^{the 3-adic valuation of n}.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 16, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2

Offset: 1

Antti Karttunen, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079, A007949, A046644, A317932.

[2^Valuation(n, 3): n in [1..100]]; // Vincenzo Librandi, Mar 19 2020

Table[2^IntegerExponent[n, 3], {n, 100}] (* Vincenzo Librandi, Mar 19 2020 *)

PARI
```
A318666(n) = 2^valuation(n,3);
```

A318666(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(3 == f[i,1], m *= 2^f[i,2])); (m); };

a(n) = 2^A007949(n).
a(n) = A046644(n)/A317932(n).
Multiplicative with a(3^e) = 2^e, a(p^e) = 1 for any other primes.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Nov 17 2022
Dirichlet g.f.: zeta(s)*(3^s-1)/(3^s-2). - Amiram Eldar, Jan 03 2023
More precise asymptotics: Sum_{k=1..n} a(k) ~ 2*n + zeta(log(2)/log(3)) * n^(log(2)/log(3)) / (2*log(2)). - Vaclav Kotesovec, Jun 25 2024

A318669 Numerators of the sequence whose Dirichlet convolution with itself yields A065769 ("Prime cascade").

Original entry on oeis.org

1, 1, 1, 7, 3, 1, 5, 25, 5, 3, 7, 7, 11, 5, 3, 363, 13, 5, 17, 21, 5, 7, 19, 25, 51, 11, 13, 35, 23, 3, 29, 1335, 7, 13, 15, 35, 31, 17, 11, 75, 37, 5, 41, 49, 15, 19, 43, 363, 115, 51, 13, 77, 47, 13, 21, 125, 17, 23, 53, 21, 59, 29, 25, 9923, 33, 7, 61, 91, 19, 15, 67, 125, 71, 31, 51, 119, 35, 11, 73, 1089, 139, 37, 79, 35, 39, 41, 23

Offset: 1

Antti Karttunen, Sep 03 2018

Multiplicative because A065769 and A317932 are.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A065769, A317932 (denominators).

up_to = 1+(2^16);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A065769(n) = { my(f=factor(n>>valuation(n,2))[, 1]~); (A003557(n) * factorback(vector(#f,i,precprime(f[i]-1)))); }; \\ Antti Karttunen, Sep 03 2018
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA065769(n)));
A318669(n) = numerator(v318669_aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A065769(n) - Sum_{d|n, d>1, d 1.

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A318666 a(n) = 2^{the 3-adic valuation of n}.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A318669 Numerators of the sequence whose Dirichlet convolution with itself yields A065769 ("Prime cascade").

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula