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.

A349826 Iterate x -> A349824(x) starting at n; if trajectory reaches a fixed point then that is a(n), if it ends in the loop (28,33) then a(n) = 28, otherwise a(n) = -1.

Original entry on oeis.org

0, 0, 2, 3, 28, 5, 28, 7, 28, 27, 28, 11, 27, 13, 28, 28, 28, 17, 28, 19, 27, 27, 30, 23, 28, 27, 30, 27, 28, 29, 30, 31, 28, 28, 28, 28, 28, 37, 28, 28, 28, 41, 28, 43, 28, 28, 28, 47, 28, 28, 28, 28, 28, 53, 28, 28, 28, 28, 28, 59, 28, 61, 28, 28, 28, 28, 28
Offset: 0

Views

Author

N. J. A. Sloane, Jan 01 2022

Keywords

Comments

It is conjectured that every trajectory eventually reaches one of the fixed points {primes union 0, 27, 30} or the loop (28, 33).

Examples

			Trajectory of 16 is 16, 32, 50, 36, 40, 44, 45, 33, 28, 33, 28, 33, 28, 33, 28, 33, 28, 33, 28, ..., ending at the loop (28, 33), so a(n) = 28.

Links

Crossrefs

Cf. A349824-A349827.

Programs

  • PARI
    a(n) = { for (k=0, oo, my (m=if (n==0, 0, my (f=factor(n)); bigomega(f)*sum(k=1, #f~, f[k,1]*f[k,2]))); if (n==28 || m==n, return (n), n=m) ) } \\ Rémy Sigrist, Jan 02 2022

Extensions

More terms from Rémy Sigrist, Jan 02 2022

A349827 Iterate x -> A349824(x) starting at n; a(n) is the greatest term in the trajectory, or -1 if the trajectory increases for ever.

Original entry on oeis.org

0, 1, 2, 3, 45, 5, 45, 7, 45, 27, 45, 11, 27, 13, 45, 50, 50, 17, 45, 19, 27, 27, 30, 23, 45, 27, 30, 27, 33, 29, 30, 31, 50, 33, 45, 45, 45, 37, 45, 50, 45, 41, 45, 43, 45, 45, 50, 47, 55, 49, 50, 51, 52, 53, 54, 55, 56, 57, 66, 59, 60, 61, 66, 63, 72, 65, 66
Offset: 0

Views

Author

N. J. A. Sloane, Jan 01 2022

Keywords

Comments

It is conjectured that every trajectory eventually reaches one of the fixed points {primes union 0, 27, 30} or the loop (28, 33).

Examples

			Trajectory of 16 is 16, 32, 50, 36, 40, 44, 45, 33, 28, 33, 28, 33, 28, 33, 28, 33, 28, 33, 28, ..., ending at the loop (28, 33), and the high-point is 50, so a(16) = 50.

Links

Crossrefs

Cf. A349824-A349826.

Programs

  • PARI
    a(n) = { my (s=[]); while (!setsearch(s, n), s=setunion(s, [n]); n=if (n==0, 0, my (f=factor(n)); bigomega(f)*sum(k=1, #f~, f[k,1]*f[k,2]))
); s[#s] } \\ Rémy Sigrist, Jan 01 2022

Extensions

More terms from Rémy Sigrist, Jan 01 2022

A349825 Number of steps when x -> A349824(x) is iterated starting at n needed to reach fixed point or 28, or -1 if trajectory increases for ever or ends in a nontrivial loop other than (28,33).

Original entry on oeis.org

0, 1, 0, 0, 9, 0, 10, 0, 8, 4, 9, 0, 3, 0, 8, 9, 8, 0, 7, 0, 1, 2, 2, 0, 6, 2, 1, 0, 0, 0, 0, 0, 7, 1, 8, 7, 5, 0, 7, 8, 4, 0, 6, 0, 3, 2, 7, 0, 9, 1, 6, 5, 6, 0, 4, 8, 7, 4, 12, 0, 10, 0, 11, 9, 12, 6, 10, 0, 10, 7, 7, 0, 11, 0, 6, 9, 8, 6, 5, 0, 7, 10, 9, 0
Offset: 0

Views

Author

N. J. A. Sloane, Jan 01 2022

Keywords

Comments

It is conjectured that every trajectory eventually reaches one of the fixed points {primes union 0, 27, 30} or the loop (28, 33).
a(n) = number of steps to reach A349826(n) (or -1).

Examples

			Trajectory of 16 is 16, 32, 50, 36, 40, 44, 45, 33, 28, 33, 28, 33, 28, 33, 28, 33, 28, 33, 28, ..., reaching low point of 28 after 8 steps, so a(16) = 8.

Links

Crossrefs

Cf. A349824-A349827.

Programs

  • PARI
    a(n) = { for (k=0, oo, my (m=if (n==0, 0, my (f=factor(n)); bigomega(f)*sum(k=1, #f~, f[k,1]*f[k,2]))); if (n==28 || m==n, return (k), n=m) ) } \\ Rémy Sigrist, Jan 02 2022

Extensions

More terms from Rémy Sigrist, Jan 02 2022

A350747 Number of iterations required to terminate trajectory mapping described in A349824.

Original entry on oeis.org

0, 1, 0, 0, 8, 0, 9, 0, 7, 4, 8, 0, 3, 0, 7, 8, 7, 0, 6, 0, 1, 2, 2, 0, 5, 2, 1, 0, 0, 0, 0, 0, 6, 0, 7, 6, 4, 0, 6, 7, 3, 0, 5, 0, 2, 1, 6, 0, 8, 1, 5, 4, 5, 0, 3, 7, 6, 3, 11, 0, 9, 0, 10, 8, 11, 5, 9, 0, 9, 6, 6, 0, 10
Offset: 0

Views

Author

Gary Detlefs, Jan 13 2022

Keywords

Comments

From Wolfdieter Lang, Feb 09 2022: (Start)
Conjecture from A349824: the iteration f: n -> A349824(n) becomes periodic for each n >= 0.
a(n) gives the number of steps from n to reach the first member of the periodic part. There are the two length 2 periods: (33,28) and (28,33). (End)
It appears that the only nonprime values of n for which a(n) = 0 are {0, 27, 28, 30, 33}.

Examples

			For n = 6, the trajectory is 6, 10, 14, 18, 24, 36, 40, 44, 45, 33, ... so a(6) = 9.
For n = 24, the trajectory is 24, 36, 40, 44, 45, 33, ... so a(24) = 5.

Crossrefs

Cf. A349824, A001414, A001222.

Programs

  • Maple
    with(numtheory): A001222:= n -> bigomega(n):
A001414:= proc(n) local e, j; e:=ifactors(n)[2]; add(e[j][1] * e[j][2],j= 1..nops(e)) end proc :
B := n-> A001414(n) * A001222(n):
g:= proc(n) if isprime(n) or n=0 or n=27 or n=28 or n=30 or n=33 then return 0 else return 1 fi end proc:
F:= proc(n) local v,i; v:=n;if n = 1 then return 1 else if g(n)=0 then return 0 else for i from 0 to 100 do v:= B(v);if  v=27 or v=28 or v=30 or v=33 then return i+1; i:=100 fi od fi fi end proc :
Seq(F(n), n=0..100)

Extensions

More terms from Jinyuan Wang, Jan 15 2022
Edited by Wolfdieter Lang, Feb 09 2022
Showing 1-4 of 4 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.