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 1-4 of 4 results.

A366892 a(n) = A336652(A163511(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 12, 3, 5, 1, 39, 12, 30, 3, 15, 5, 7, 1, 120, 39, 155, 12, 90, 30, 56, 3, 60, 15, 35, 5, 21, 7, 11, 1, 363, 120, 780, 39, 465, 155, 399, 12, 360, 90, 280, 30, 168, 56, 132, 3, 195, 60, 210, 15, 105, 35, 77, 5, 84, 21, 55, 7, 33, 11, 13, 1, 1092, 363, 3905, 120, 2340, 780, 2800, 39, 1860, 465, 1995
Offset: 0

Views

Author

Antti Karttunen, Nov 04 2023

Keywords

Links

Crossrefs

Cf. A163511, A336652, A366893 (rgs-transform).
Cf. also A324186.

Programs

  • PARI
    A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A336652(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],1,-1+(((f[i,1]^(1+f[i,2]))-1) / (f[i,1]-1)))));
A366892(n) = A336652(A163511(n));

A336651 Odd part of n divided by its largest squarefree divisor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 3, 5, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Jul 30 2020

Keywords

Comments

The name can be parsed either as "Odd part of {n divided by its largest squarefree divisor}" or "Odd part of n, divided by its largest squarefree divisor". Because A000265 and A003557 commute, both interpretations yield equal results.

Links

Crossrefs

Cf. A000203, A000265, A003557, A007947, A336649, A336652.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := p^(e-1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)
  • PARI
    A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };

Formula

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = p^(e-1).
a(n) = A003557(A000265(n)) = A000265(A003557(n)).
a(n) = A000265(n) / A204455(n).
A000203(a(n)) = A336649(n).
Dirichlet g.f.: (1 - 1/(2^s-1)^2) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^s). - Amiram Eldar, Sep 14 2023

A336649 Sum of divisors of A336651(n) (odd part of n divided by its largest squarefree divisor).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 6, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 6, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 6, 1, 1, 1, 1, 1, 40, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 8, 4, 6, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Jul 30 2020

Keywords

Links

Crossrefs

Cf. A000203, A000265, A204455, A335341, A336651, A336652.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^e - 1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)
  • PARI
    A336649(n) = { my(f=factor(n)); prod(i=1, #f~, if((2==f[i,1])||(1==f[i,2]),1,(((f[i,1]^(f[i,2]))-1) / (f[i,1]-1)))); };
  • PARI
    A000265(n) = (n>>valuation(n,2));
A335341(n) = if(1==n,n,sigma(n/factorback(factorint(n)[, 1])));
A336649(n) = A335341(A000265(n));

Formula

Multiplicative with a(2^e) = 1, a(p^1) = 1 and a(p^e) = (p^e - 1)/(p-1) if e > 1.
a(n) = A000203(A336651(n)) = A335341(A000265(n)).
a(n) = A336652(n) / A204455(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * (1 - 1/(1-2^s+2^(2*s-1))) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Dec 18 2023

A366893 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366892(i) = A366892(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 31, 16, 32, 1, 33, 17, 34, 9, 35, 18, 36, 5, 37, 19, 38, 10, 39, 20, 40, 3, 41, 21, 42, 11, 43, 22, 44, 6, 45, 23
Offset: 0

Views

Author

Antti Karttunen, Nov 04 2023

Keywords

Comments

Restricted growth sequence transform of A366892.

Links

Crossrefs

Cf. A163511, A336652, A366892.
Cf. also A366806.
Differs from A003602 for the first time at n=121, where a(121) = 24, while A003602(121) = 61.

Programs

  • PARI
    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; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A336652(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],1,-1+(((f[i,1]^(1+f[i,2]))-1) / (f[i,1]-1)))));
A366892(n) = A336652(A163511(n));
v366893 = rgs_transform(vector(1+up_to,n,A366892(n-1)));
A366893(n) = v366893[1+n];
Showing 1-4 of 4 results.

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