Johannes M.V.A. Koelman - cp's OEIS frontend

User: Johannes M.V.A. Koelman

Johannes M.V.A. Koelman has authored 3 sequences.

A380609 Primes a single step away from a cycle under the mapping p-> gpf(2*p+1).

2, 17, 31, 37, 67, 71, 73, 97, 103, 137, 149, 157, 181, 199, 211, 227, 241, 269, 283, 313, 337, 367, 379, 409, 487, 541, 563, 577, 587, 607, 617, 643, 661, 769, 787, 857, 877, 907, 929, 937, 977, 997, 1039, 1093, 1151, 1187, 1237, 1453, 1543, 1567, 1579, 1621

Offset: 1

Johannes M.V.A. Koelman, Jan 28 2025

nonn

The cycle that gets entered consists of the primes in A287865. It appears that the mapping p -> gpf(2*p+1) produces no other cycles.

Conjecture: under repeated mapping all primes ultimately enter the same cycle.

			Prime 2 is in the sequence as it maps to 5. And so is 17 as it maps to 7.  The primes 3, 5, 7, 11, 13, 19, 23 and 47 are not included, as they are part of the cycle itself (and hence considered zero iterations away from the cycle).

Cf. A006530, A076565, A287865.

gpf:= n -> max(numtheory:-factorset(n)):
filter:= proc(n) local S,t,x;
  t:= gpf(2*n+1);
  if t = n then return false fi;
  S:= {n,t};
  x:= t;
  do
    x:= gpf(2*x+1);
    if member(x,S) then return (x = t) fi;
    S:= S union {x};
  od;
end proc:
select(filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Feb 03 2025

A380312 Primes not reaching 3 under iterations of p -> gpf(2*p-1).

Original entry on oeis.org

19, 29, 37, 67, 73, 101, 131, 167, 181, 197, 211, 241, 251, 257, 317, 389, 421, 463, 479, 503, 523, 599, 643, 653, 691, 719, 739, 811, 827, 859, 887, 907, 919, 941, 983, 1039, 1061, 1069, 1109, 1117, 1217, 1277, 1283, 1289, 1307, 1361, 1381, 1427, 1429, 1499

Offset: 1

Johannes M.V.A. Koelman, Jan 20 2025

It appears that this is the primes reaching 19 under iterations of p -> gpf(2*p-1).

Conjecture: a(n) ~ k*n log n for some constant k. Perhaps k ≈ 4.51. - Charles R Greathouse IV, Jan 24 2025

			19, 37, 73 and 29 are in the sequence as they form a loop under the iteration.

Cf. A006530, A023583.

\\ This will loop forever if it hits a loop other than 3 or 19, but if it returns the result is correct.
gpf(n)=my(f=factor(n)[,1]); f[#f]
is(p)=while(p>28, p=gpf(2*p-1)); p==19 \\ Charles R Greathouse IV, Jan 24 2025

A349873 Smallest odd value such that any Collatz trajectory in which it occurs contains exactly n odd values other than '1'.

Original entry on oeis.org

21, 3, 69, 45, 15, 9, 51, 33, 87, 57, 39, 105, 135, 363, 123, 339, 219, 159, 393, 519, 681, 897, 603, 111, 297, 1581, 1053, 351, 933, 621, 207, 549, 183, 243, 645, 429, 285, 189, 63, 165, 27, 147, 195, 129, 171, 231, 609, 411, 543, 1449, 975, 327, 873, 1185, 1527, 1017

Offset: 1

Johannes M.V.A. Koelman, Dec 03 2021

nonn

a(n) necessarily is the first odd term in any Collatz trajectory in which it occurs.

			a(1)=21 as 21 occurs solely in Collatz trajectories starting with 21*2^k, and these trajectories all contain one single odd value other than 1. No value smaller than 21 satisfies these requirements. In particular, a(1) does not equal 5 since 5 is part of Collatz trajectories that contain multiple odd values other than 1 (e.g., ...,13,40,20,10,5,16,8,4,2,1).
a(2)=3 as 3 occurs solely in Collatz trajectories starting with 3*2^k, and these trajectories all contain exactly two odd values other than 1 (namely 3 and 5).

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A075677, A075680, A006667.

All terms are in A016945.

oddsteps(n)={my(s=0); while(n!=1, if(n%2,n=(3*n+1);s++); n/=2); s}
a(n)={forstep(k=3, oo, 6, if(oddsteps(k)==n, return(k)))} \\ Andrew Howroyd, Dec 19 2021

oddsteps(n)=my(s); while(n>1, n+=n>>1+1; if(!bitand(n,1), n >>= valuation(n,2)); s++); s
first(n)=my(v=vector(n),r=n,t); forstep(k=3,oo,2, t=oddsteps(k); if(t<=n && v[t]==0, v[t]=k; if(r-- == 0, return(v)))) \\ Charles R Greathouse IV, Dec 22 2021

a(n) mod 6 = 3 for all n>0. The odd multiples of 3 form the 'Garden-of-Eden' set (terms without a predecessor) under iterations of the reduced Collatz function A075677.

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User: Johannes M.V.A. Koelman

A380609 Primes a single step away from a cycle under the mapping p-> gpf(2*p+1).

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Maple

A380312 Primes not reaching 3 under iterations of p -> gpf(2*p-1).

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PARI

A349873 Smallest odd value such that any Collatz trajectory in which it occurs contains exactly n odd values other than '1'.

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PARI

PARI

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