A138754 - OEIS

A138754

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

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

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OFFSET

1,2

COMMENTS

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).

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

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

FORMULA

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

EXAMPLE

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).

MATHEMATICA

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

PROG

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