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 1-2 of 2 results.

A334861 a(n) = A329697(n) + A331410(n).

Original entry on oeis.org

0, 0, 2, 0, 3, 2, 3, 0, 4, 3, 4, 2, 4, 3, 5, 0, 4, 4, 6, 3, 5, 4, 5, 2, 6, 4, 6, 3, 7, 5, 4, 0, 6, 4, 6, 4, 7, 6, 6, 3, 5, 5, 7, 4, 7, 5, 6, 2, 6, 6, 6, 4, 7, 6, 7, 3, 8, 7, 8, 5, 5, 4, 7, 0, 7, 6, 8, 4, 7, 6, 7, 4, 8, 7, 8, 6, 7, 6, 7, 3, 8, 5, 6, 5, 7, 7, 9, 4, 8, 7, 7, 5, 6, 6, 9, 2, 5, 6, 8, 6, 8, 6, 6, 4, 8
Offset: 1

Views

Author

Antti Karttunen, May 14 2020

Keywords

Comments

Completely additive because A329697 and A331410 are. No 1's occur as terms.

Links

Crossrefs

Cf. A000079 (positions of zeros), A329697, A331410, A334862.

Programs

  • PARI
    A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
A334861(n) = (A329697(n)+A331410(n));
\\ Or alternatively as:
A334861(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(2+A329697(f[k,1]-1)+A331410(f[k,1]+1)))); };

Formula

a(n) = A329697(n) + A331410(n).
a(2) = 0, a(p) = 2+A329697(p-1)+A331410(p+1) for odd primes p, a(m*n) = a(m)+a(n), if m,n > 1.

A334097 a(n) is the exponent of the eventual power of 2 reached when starting from n and using the map k -> k + k/p, where p can be any odd prime factor of k, for example, the largest.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 6, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 7, 8, 7, 7, 7, 8, 7, 8, 7, 8, 8, 7, 7, 7, 7, 8, 7, 7, 7, 8, 8, 8, 8, 7, 7, 8
Offset: 1

Views

Author

Antti Karttunen, Apr 29 2020

Keywords

Links

Crossrefs

Cf. A007814, A052126, A078701, A209229, A331410, A334098, A334862.
Cf. also A064415 (analogous sequence when using the map k -> k - k/p).

Programs

  • Mathematica
    Array[Log2@ NestWhile[# + #/FactorInteger[#][[-1, 1]] &, #, !IntegerQ@ Log2@ # &] &, 105] (* Michael De Vlieger, Apr 30 2020 *)
  • PARI
    A334097(n) = if(!bitand(n,n-1),valuation(n,2),my(f=factor(n)[, 1]); A334097(n+(n/f[2-(n%2)])));
  • PARI
    A334097(n) = if(!bitand(n,n-1),valuation(n,2),A334097(n+(n/vecmax(factor(n)[, 1]))));
  • PARI
    A334097(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],f[k,2],f[k,2]*A334097(1+f[k,1]))); };

Formula

Totally additive sequence: a(2) = 1, a(p) = a(p+1) for odd primes p, a(m*n) = a(m)+a(n) for m, n > 1.
If A209229(n) == 1, a(n) = A007814(n), otherwise a(n) = a(n+A052126(n)), or equally, a(n) = a(n+(n/A078701(n))).
a(n) = A331410(n) + A334098(n) = A334862(n) + A064415(n).
Showing 1-2 of 2 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.