A138750 -id:A138750 - cp's OEIS frontend

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).

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

A138751 a(n) = nextprime( p(n)/2 if p(n)=2 (mod 3), 2p(n) else ) = A007918( A138750( A000040( n ))).

Original entry on oeis.org

2, 7, 3, 17, 7, 29, 11, 41, 13, 17, 67, 79, 23, 89, 29, 29, 31, 127, 137, 37, 149, 163, 43, 47, 197, 53, 211, 59, 223, 59, 257, 67, 71, 281, 79, 307, 317, 331, 89, 89, 97, 367, 97, 389, 101, 401, 431, 449, 127, 461, 127, 127, 487, 127, 131, 137, 137, 547, 557, 149

Offset: 1

M. F. Hasler, Mar 28 2008

Composing the map A138750 with A007918 to the left and restricting it to the primes makes it a mapping from primes into primes which is a natural generalization of the Collatz problem to primes. (Looking at parity would not be interesting for primes, so using "mod 3" is the simplest nontrivial generalization.)
The only even prime p=2 is the only fixed point of this map and all odd primes seem to end up in the loop 7 -> 17 -> 11 -> 7, after a number of steps given in A138752.
The sequence A124123 lists the primes which do not occur in the present sequence.
See A138750 for further information.

			a(1) = nextprime(2/2) = 2, a(2) = nextprime(2*3) = 7, a(3) = nextprime(5/2) = 7.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Georges Brougnard, Definition of GB-sequences.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A124123, A007918, A138750, A138752-A138753.

A138751[n_]:=With[{p=Prime[n]},NextPrime[If[Mod[p,3]==2,p/2,2p]]];Array[A138751,100] (* Paolo Xausa, Jul 28 2023 *)

A138751(n) = { n=prime(n); nextprime( if( n%3==2, ceil(n/2), 2*n ))}

a(n) = A007918(A138750(A000040(n))).

A138752 Number of iterations before prime(n) reaches 7 or 2 under x -> A007918(A138750(x)).

Original entry on oeis.org

0, 1, 2, 0, 1, 4, 2, 7, 5, 3, 20, 16, 6, 6, 4, 4, 21, 23, 19, 17, 17, 15, 7, 5, 7, 5, 28, 22, 26, 22, 22, 20, 18, 18, 16, 20, 16, 14, 6, 6, 8, 59, 8, 8, 6, 29, 27, 25, 23, 25, 23, 23, 27, 23, 21, 19, 19, 21, 19, 17, 19, 17, 19, 17, 17, 15, 13, 11, 9, 11, 9, 60, 58, 54, 11, 9, 7, 30, 28

Offset: 1

M. F. Hasler, Mar 28 2008

As explained in A138751, the map x->A007918(A138750(x)) is a natural generalization of the Collatz map to primes.
The only even prime p=2 is the only fixed point of this map, and all odd primes seem to end up in the loop 7 -> 17 -> 11 -> 7, after a number of steps given in the present sequence.
(It might have been more natural to count the steps until a number is reached for the second time. Depending on which number among {2,7,11,17} is reached first, this would increase the value of a(n) by 1,3,2 resp. 1.)

			a(1)=a(4)=0 since prime(1)=2 and prime(4)=7 are by definition the values at which counting ends.
a(primepi(4499221))=63337 according to G. Brougnard, cf. Link.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Georges Brougnard, Trajectory of 4499221.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A124123, A138750, A138751, A138753.

A138752[n_]:=Length[NestWhileList[NextPrime[If[Mod[#,3]==2,#/2,2#]]&,Prime[n],#!=2&&#!=7&]]-1;Array[A138752,100] (* Paolo Xausa, Jul 28 2023 *)

A138752(n,c=0) = { if( n==1 & 7==n=prime(n), 0, until( 7==n=nextprime( if( n%3==2, ceil(n/2), 2*n )),c++);c)}

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))

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

Original entry on oeis.org

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)).

A138754 a(n) = PrimePi(A138751(n)) - a variation of the Collatz (3n+1) map.

Original entry on oeis.org

1, 4, 2, 7, 4, 10, 5, 13, 6, 7, 19, 22, 9, 24, 10, 10, 11, 31, 33, 12, 35, 38, 14, 15, 45, 16, 47, 17, 48, 17, 55, 19, 20, 60, 22, 63, 66, 67, 24, 24, 25, 73, 25, 77, 26, 79, 83, 87, 31, 89, 31, 31, 93, 31, 32, 33, 33, 101, 102, 35, 104, 35, 113, 37, 115, 38, 122, 123, 41, 126

Offset: 1

M. F. Hasler, Apr 01 2008

This map is a variation of the Collatz (or 3n+1) map:
Instead of considering the parity of the number, we look at prime(n) mod 3 to decide if this prime should be halved or doubled, before going to the next prime (A007918) and finally back to the positive integers via PrimePi (A000720).
Exactly as for the Collatz (3n+1) map (defined on nonnegative integers), the first element for which it is defined is its only fixed point, and all other starting values seem to end up in a cycle of length 3, here: 4 -> 7 -> 5 -> 4.
Except for p=3, no prime yields a prime result under the map A138750 (as can be seen using p=6k+1 or p=6k-1). Therefore instead of applying primepi() after nextprime(), one could also simply use 1+primepi().
The prime p=3 is also the only case where n == 2 (mod 3) is not equivalent to n != 1 (mod 3). It might have been a better choice to define A138750(x)=2x if x == 1 (mod 3), ceiling(x/2) otherwise. But since here it makes only a difference for p=3, we use the original definition (cf. A124123).

			a(4) = 7 since prime(4) = 7 == 1 (mod 3), thus A138750(7) = 2*7 = 14, nextprime(14) = 17, PrimePi(17) = 7 (i.e., 17 is the 7th prime).
a(5) = 4 since prime(5) = 11 == 2 (mod 3), thus A138750(11) = ceiling(11/2) = 6, nextprime(6) = 7, PrimePi(7) = 4 (i.e., 7 is the 4th prime).

Cf. A124123, A138750-A138753, A000040, A000720, A007918.

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

A138754(n)=primepi(A138751(n)) /* see there */

a(n) = A000720(A138751(n)) = A000720(A007918(A138750(A000040(n)))).

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

Showing 1-8 of 8 results.

cp's OEIS Frontend

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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A138751 a(n) = nextprime( p(n)/2 if p(n)=2 (mod 3), 2p(n) else ) = A007918( A138750( A000040( n ))).

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A138752 Number of iterations before prime(n) reaches 7 or 2 under x -> A007918(A138750(x)).

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

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A138754 a(n) = PrimePi(A138751(n)) - a variation of the Collatz (3n+1) map.

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

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