A138754 -id:A138754 - cp's OEIS frontend

A138753 Number of iterations of A138754 before reaching a number for the second time, when starting with n.

1, 4, 5, 3, 3, 5, 3, 8, 6, 4, 21, 17, 7, 7, 5, 5, 22, 24, 20, 18, 18, 16, 8, 6, 8, 6, 29, 23, 27, 23, 23, 21, 19, 19, 17, 21, 17, 15, 7, 7, 9, 60, 9, 9, 7, 30, 28, 26, 24, 26, 24, 24, 28, 24, 22, 20, 20, 22, 20, 18, 20, 18, 20, 18, 18, 16, 14, 12, 10, 12, 10, 61, 59, 55, 12, 10, 8, 31

Offset: 1

M. F. Hasler, Apr 01 2008

nonn

This is a variation of A138752, giving the number of iterations of A138754 needed to get any number for the second time, while A138752 stops counting somehow arbitrarily at 1=primepi(2) or 4=primepi(7).
The map A138754 is a variation of the Collatz map where parity of the integers is replaced by p mod 3 for the primes.
For the Collatz map, we have the only fixed point 0=f(0) and all other numbers seem to end up in the cycle 1->4->2->1.
Here the only fixed point is 1=A138754(1) and all other numbers seem to end up in the cycle 4 -> 7 -> 5 -> 4 (corresponding to primes 7 -> 17 -> 11 -> 7).
Depending on which number among primepi({2,7,11,17}) is reached first, A138753(n) = A138752(n)+1,+3,+2 resp. +1. (A138752(n) is the length of the so-called GB-sequence starting with prime(n).)

			a(1)=1 since after 1 step we find 1 again.
a(4)=3 since 4 -> 7 -> 5 -> 4 under A138754.

Paolo Xausa, Table of n, a(n) for n = 1..1459 (terms 1..500 from M. F. Hasler)
Georges Brougnard, Definition of GB-sequences.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A124123, A006577, A171938, A138756 (record values/indices).

Cf. A138750, A138751, A138752, A138754.

A138754[n_]:=A138754[n]=With[{p=Prime[n]},PrimePi[NextPrime[If[Mod[p,3]==2,p/2,2p]]]];
A138753[n_]:=Length[NestWhileList[A138754,n,UnsameQ,{1,4}]]-1;
Array[A138753,100] (* Paolo Xausa, Jul 28 2023*)

A138753(n,c=0,t=[1,1,1]) = { until( t[c++%3+1]==n=A138754(n), t[c%3+1]=n); c}

a(n) = min { k>0 | A138754^k(n) = A138754^m(n) for some m>=0, m

If n is not in {1,4,5,7}, then a(n) = 1+a(A138754(n)).

A124123 Primes not of the form nextprime(f(p)) with p prime, where f(p)=p/2 if p=2 (mod 3), f(p)=2p otherwise (cf. A138750).

Original entry on oeis.org

5, 19, 61, 73, 83, 103, 107, 109, 113, 139, 151, 167, 173, 191, 199, 229, 269, 271, 277, 313, 337, 349, 359, 379, 397, 439, 463, 503, 523, 563, 571, 601, 607, 619, 733, 773, 823, 827, 829, 859, 883, 887, 911, 971, 983, 997, 1013, 1031, 1063, 1091, 1093, 1103

Offset: 1

Jacques Tramu, Dec 13 2006

These are the primes which cannot be part of a gb-sequence (except as seed).
Is this sequence finite or infinite?
From M. F. Hasler, Mar 27 2008: (Start)
The last comment above probably refers not to this sequence but to the "gb-sequences" themselves, e.g., the one starting with 4499221 which reaches a peak of approximately 10^110, cf. Formula and Links.
The function f(p)=p/2 if p == 2 (mod 3), f(p)=2p otherwise, yields a half-integer for primes p=6k-1 and an even number for primes p=6k+1; in all cases nextprime(f(p)) is defined without ambiguity: f(p) will never be equal to a prime.
This sequence lists primes p' not in the range of the map p -> nextprime(f(p)), defined on the primes.
Equivalently, p' is listed iff: (i) no even number between p' and the next lower prime is of the form 2p with p=0 or p == 1 (mod 3), AND (ii) no half-integer between p' and the next lower prime is of the form p/2 with p == 2 (mod 3) and p prime (in both conditions).
This characterization allows easy computation of the sequence, cf. PARI code.
Experimentally, it does not appear that this sequence is finite. Instead, its (local) density within the primes seems to increase, from roughly 25% for the first terms to about 50% at 10^30. (End)
The function f is discussed in A138750. Composed with the nextprime function and restricted to the primes (cf. A138751), it yields a ("natural") variant of the Collatz function on the set of the primes, with (mod 2) replaced by (mod 3). The gb-sequences are the orbits under that function. - M. F. Hasler, Nov 18 2018

			Example: a(1) = 5 because there is no prime gb(n) such that gb(n+1) = 5.

Communication paper by Georges Brougnard.

Vincenzo Librandi, Table of n, a(n) for n = 1..6470
Georges Brougnard, Definition of GB-sequences.
Georges Brougnard, GB-sequence of length 96, obtained for gb[0]=1381.
Georges Brougnard, GB-sequence of length 63337, obtained for gb[0]=4499221.

Cf. A007918 (nextprime), A138750 (function f), A138751, A138752, A138753, A138754.

lim = PrimePi[1000]; f[p_ /; Mod[p, 3] == 2] := p/2; f[p_] := 2*p; Complement[Prime[Range[lim]], Table[ NextPrime[ f[Prime[k]]], {k, 1, 2*lim}]] (* Jean-François Alcover, Sep 20 2011 *)

{forprime( p=3,10^3, for( i=precprime(p-1)+1,p, (2*i)%3==0 & isprime(2*i-1) & next(2); i%2==0 & ( i/2 )%3!=2 & isprime( i/2 ) & next(2)); print1( p", " ))}
nextA124123(p)={ while( p=nextprime(p+1), for( i=precprime(p-1)+1,p, (2*i)%3==0 & isprime(2*i-1) & next(2); i%2==0 & ( i/2 )%3!=2 & isprime( i/2 ) & next(2)); return( p )) }
t=2;vector(200,i,t=nextA124123(t)) \\ 60% of the first 200 terms are in 1+3Z:
t=[0,0];vector(#%,i,t[%[i]%3]++);t \\ yields [120, 80]
t=10^11;vector(200,i,t=nextA124123(t)) \\ exactly 50% of these terms are in 1+3Z:
t=[0,0];vector(#%,i,t[%[i]%3]++);t \\ yields [100, 100]
t=10^30;vector(200,i,t=nextA124123(t+1));t-10^30 \\ yields 31773 = distance of 200th term beyond 10^30
t=10^30;vector(200,i,t=nextprime(t+1));(t-1e30)/% \\ yields 0.52..., approx. local density in the primes. (End)

Complement of A007918(A138750(A000040)) = nextprime(f({primes})).
Recurrence for a gb-sequence starting with gb(0) = a prime > 2 (the seed):
| If gb(n) = 2 (mod 3) then gb(n+1) := least prime > gb(n)/2;
| otherwise gb(n+1) := least prime > gb(n)*2.
A gb-sequence of length L ends in the loop 7, 17, 11, 7, ... ; gb(L-1) = 7.

Edited by M. F. Hasler, Mar 27 2008, Nov 18 2018

A138756 Indices of record values in A138753 (a "prime" variation of the Collatz (3n+1) problem).

Original entry on oeis.org

1, 2, 3, 8, 11, 17, 18, 27, 42, 72, 125, 219, 221, 401, 515, 556, 754, 841, 1146

Offset: 1

M. F. Hasler, Apr 01 2008

"Indices of ..." is equivalent to "starting values for ..."

Cf. A124123, A138750-A138754, A171938, A006877 (analog for Collatz problem).

A138754[n_]:=A138754[n]=With[{p=Prime[n]},PrimePi[NextPrime[If[Mod[p,3]==2,p/2,2p]]]];
A138753[n_]:=Length[NestWhileList[A138754,n,UnsameQ,{1,4}]]-1;
A138756list[upto_]:=Module[{v,r=0},Table[If[(v=A138753[n])>r,r=v;n,Nothing],{n,upto}]];
A138756list[500] (* Paolo Xausa, Jul 30 2023 *)

m=0; for( i=1,#A138753, A138753[i] > m || next; m=A138753[i]; print1(i", "))

a(n) = min { k | A138753(k) = A171938(n) }

Equals { m | A138753(k) < A138753(m) for all k

a(15)-a(19) from Paolo Xausa, Jul 30 2023

A171938 Record values in A138753 (a "prime" variation of the Collatz (3n+1) problem).

Original entry on oeis.org

1, 4, 5, 8, 21, 22, 24, 29, 60, 61, 72, 73, 97, 100, 184, 216, 239, 451, 469

Offset: 1

M. F. Hasler, Apr 01 2008

Cf. A124123, A138750-A138754, A138756, A006878 (analog for Collatz problem).

A138754[n_]:=A138754[n]=With[{p=Prime[n]},PrimePi[NextPrime[If[Mod[p,3]==2,p/2,2p]]]];
A138753[n_]:=Length[NestWhileList[A138754,n,UnsameQ,{1,4}]]-1;
A171938list[upto_]:=Module[{v,r=0},Table[If[(v=A138753[n])>r,r=v,Nothing],{n,upto}]];
A171938list[500] (* Paolo Xausa, Jul 29 2023 *)

m=0; for( i=1,#A138753, A138753[i] > m & print1( m=A138753[i],", "))

A171938(n) = A138753(A138756(n)).

A171938 = { A138753(m) | A138753(k) < A138753(m) for all k

Originally submitted as A138755, but mislaid by Editor-in-Chief; renumbered and added to OEIS, Oct 24 2010

a(15)-a(19) from Paolo Xausa, Jul 29 2023

A138757 a(n) = A007918(A138750(n)), that is, least prime > n/2 if n=2 (mod 3), > 2n otherwise.

Original entry on oeis.org

2, 2, 2, 7, 11, 3, 13, 17, 5, 19, 23, 7, 29, 29, 7, 31, 37, 11, 37, 41, 11, 43, 47, 13, 53, 53, 13, 59, 59, 17, 61, 67, 17, 67, 71, 19, 73, 79, 19, 79, 83, 23, 89, 89, 23, 97, 97, 29, 97, 101, 29, 103, 107, 29, 109, 113, 29, 127, 127, 31, 127, 127, 31, 127

Offset: 0

M. F. Hasler, Apr 04 2008

This can be considered as an analog of the Collatz (or 3n+1) map on the set of primes, see A138751 and A138754 for details.

Numbers 0,1,2 go immediately to the unique fixed point 2, all others end up in the cycle 7 -> 17 -> 11 -> 7, after a number of iterations given by A138753(A138757(n))-1 (= A138753(n)-2 if n is prime).

			a(7) = 17 since 7 = 1 (mod 3), thus A138750(7) = 2*7 = 14, nextprime(14) = 17.
a(11) = 7 since 11 = 2 (mod 3), thus A138750(11) = ceiling(11/2) = 6, nextprime(6) = 7.

Georges Brougnard, Definition of GB-sequences.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A124123, A138750, A138751, A138752, A138753, A138754, A138755, A138756, A007918.

np1[n_]:=Module[{x=Ceiling[n/2]},If[PrimeQ[x],x,NextPrime[x]]]; np2[n_]:= Module[{x=2n},If[PrimeQ[x],x,NextPrime[x]]]; Table[If[Mod[n,3]==2, np1[n], np2[n]],{n,0,70}] (* Harvey P. Dale, Jul 10 2013 *)

A138757(n)=nextprime(if(n%3==2,(n+1)\2,2*n))

a(n) = A007918(A138750(n)).

For p prime, a(p) = A138751(A000720(p))

A318759 Numbers x whose trajectory reaches 1 under recursive applications of the map x -> x/3 if x == 0 (mod 3), x -> (4x+2)/3 if x == 1 (mod 3), x -> (4x+1)/3 if x == 2 (mod 3).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 13, 18, 20, 27, 29, 36, 39, 40, 54, 60, 65, 81, 87, 108, 109, 117, 120, 121, 136, 146, 162, 180, 182, 195, 197, 243, 245, 261, 263, 272, 324, 327, 328, 332, 351, 360, 363

Offset: 1

Jack Warren, Sep 02 2018

nonn

Cf. A014682, A126241, A138754.

Showing 1-6 of 6 results.

cp's OEIS Frontend

A138753 Number of iterations of A138754 before reaching a number for the second time, when starting with n.

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A124123 Primes not of the form nextprime(f(p)) with p prime, where f(p)=p/2 if p=2 (mod 3), f(p)=2p otherwise (cf. A138750).

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A138756 Indices of record values in A138753 (a "prime" variation of the Collatz (3n+1) problem).

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A171938 Record values in A138753 (a "prime" variation of the Collatz (3n+1) problem).

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A138757 a(n) = A007918(A138750(n)), that is, least prime > n/2 if n=2 (mod 3), > 2n otherwise.

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A318759 Numbers x whose trajectory reaches 1 under recursive applications of the map x -> x/3 if x == 0 (mod 3), x -> (4*x+2)/3 if x == 1 (mod 3), x -> (4*x+1)/3 if x == 2 (mod 3).

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A318759 Numbers x whose trajectory reaches 1 under recursive applications of the map x -> x/3 if x == 0 (mod 3), x -> (4x+2)/3 if x == 1 (mod 3), x -> (4x+1)/3 if x == 2 (mod 3).