cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A318667 Numerators of the sequence whose Dirichlet convolution with itself yields A318307, which is multiplicative with A318307(p^e) = 2^A002487(e).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, -43, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, -5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 31, 1, 1, 1, 1, 1, 1, 1, 3, 1
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2018

Keywords

Comments

Multiplicative because A318307 and A317934 are.

Links

Crossrefs

Cf. A318307, A317934 (denominators).

Programs

  • PARI
    up_to = 1+(2^16);
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&&dA002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318307(n) = factorback(apply(e -> 2^A002487(e),factor(n)[,2]));
v318667_aux = DirSqrt(vector(up_to, n, A318307(n)));
A318667(n) = numerator(v318667_aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A318307(n) - Sum_{d|n, d>1, d 1.

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

Views

Author

Antti Karttunen, Aug 29 2018

Keywords

Links

Crossrefs

Cf. A002487, A077761, A318307.
Cf. also A046644.

Programs

  • PARI
    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]));
  • Python
    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

Formula

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

A318316 Multiplicative with a(p^e) = 2^A007306(e).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 16, 4, 4, 8, 8, 2, 8, 2, 16, 4, 4, 4, 16, 2, 4, 4, 16, 2, 8, 2, 8, 8, 4, 2, 16, 4, 8, 4, 8, 2, 16, 4, 16, 4, 4, 2, 16, 2, 4, 8, 32, 4, 8, 2, 8, 4, 8, 2, 32, 2, 4, 8, 8, 4, 8, 2, 16, 8, 4, 2, 16, 4, 4, 4, 16, 2, 16, 4, 8, 4, 4, 4, 32, 2, 8, 8, 16, 2, 8, 2, 16, 8
Offset: 1

Views

Author

Antti Karttunen, Aug 31 2018

Keywords

Links

Crossrefs

Cf. A007306, A318307, A318322.

Programs

  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A007306(n) = if(!n,1,A002487(n+n-1));
A318316(n) = factorback(apply(e -> 2^A007306(e),factor(n)[,2]));
  • Python
    from functools import reduce
from sympy import factorint
def A318316(n): return 1<Chai Wah Wu, May 18 2023

Formula

a(n) = 2^A318322(n).
a(n) = A318307(A003557(n^2)) = A318307(A003557(n))*A318307(n).

A318470 Multiplicative with a(p^e) = A260443(e).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 12, 3, 4, 6, 6, 2, 8, 2, 18, 4, 4, 4, 9, 2, 4, 4, 12, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 12, 4, 12, 4, 4, 2, 12, 2, 4, 6, 15, 4, 8, 2, 6, 4, 8, 2, 18, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 12, 2, 12, 4, 6, 4, 4, 4, 36, 2, 6, 6, 9, 2, 8, 2, 12, 8
Offset: 1

Views

Author

Antti Karttunen, Aug 30 2018

Keywords

Links

Crossrefs

Cf. A260443, A318306, A318307.
Differs from A293442 for the first time at n=32, where a(32) = 18, while A293442(32) = 10.
Cf. also A293216, A293217, A293443, A318469.

Programs

Formula

For all n >= 1, A001222(a(n)) = A318306(n).

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

Views

Author

Antti Karttunen, Aug 30 2018

Keywords

Comments

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

Links

Crossrefs

Cf. A002487, A261179, A318510.
Cf. also A318307.

Programs

  • PARI
    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); };
  • Python
    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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.