A073799 Numbers that begin a run of consecutive integers k such that PrimePi(k) divides 2^k.

2, 7, 19, 53, 131, 311, 719, 1619, 3671, 8161, 17863, 38873, 84017, 180503, 386093, 821641, 1742537, 3681131, 7754077, 16290047, 34136029, 71378569, 148948139, 310248241, 645155197, 1339484197, 2777105129, 5750079047, 11891268401, 24563311309, 50685770167, 104484802057, 215187847711

Offset

1,1

Comments

It seems that each term is a bit larger than twice the previous one.

Runs have lengths 3, 4, 4, 6, 6, 2, 8, 2, 2, 6, 18, 18, 30, 8, 24, 6, 2, 18, ..., respectively.

From Chai Wah Wu, Jan 27 2020: (Start)

Theorem: a(1) = 2 and a(n) = A033844(n) for n > 1. For n > 1, the length of the n-th run is prime(2^n+1)-prime(2^n) = A051439(n)-A033844(n) = A074325(n).

Proof: Let r > 1. If p = prime(2^r), then primepi(p) = 2^r.

primepi(p-1) = 2^r - 1. Since r > 1, 2^r - 1 > 2 and odd and thus does not divide any power of 2.

In addition 2^r < p and thus divides 2^p. This means that p is a term. Let q be such that p < q < prime(2^r+1). Then primepi(q) = 2^r and divides 2^q. Since primepi(q-1) = 2^r and divides 2^(q-1), this means that q does not start a run and thus is not a term.

Let w be such that prime(2^r+1) <= w < prime(2^(r+1)). Then 2^r + 1 <= primepi(w) < 2^(r+1) and does not divide any power of 2. This means that w is not a term.

(End)

Links

Chai Wah Wu, Table of n, a(n) for n = 1..57

Formula

Solutions to 2^(x-1) mod PrimePi(x-1) > 0 but 2^x mod PrimePi(x) = 0.

a(n) = A033844(n) for n > 1. - Chai Wah Wu, Jan 27 2020

Mathematica

aQ[k_] := Divisible[2^k, PrimePi[k]]; s = {}; len = {}; n = 2; While[Length[s] < 10, While[! aQ[n], n++]; n1 = n; While[aQ[n], n++]; If[n > n1, AppendTo[s, n1]; AppendTo[len, n - n1]]; n++]; s (* Amiram Eldar, Dec 11 2018 *)

Prog

(Python)
from sympy import prime
def A073799(n):
    return 2 if n == 1 else prime(2**n) # Chai Wah Wu, Jan 27 2020
(PARI) a(n) = if(n==1, 2, prime(2^n)); \\ Jinyuan Wang, Mar 01 2020

Crossrefs

Cf. A000079, A000720, A015910, A062173, A064367, A073797, A073798.

Cf. A033844. - R. J. Mathar, Sep 23 2008

Cf. A051439, A074325.

Sequence in context: A099484 A018030 A051354 * A040016 A145519 A030224

Adjacent sequences: A073796 A073797 A073798 * A073800 A073801 A073802

Keyword

nonn

Author

Labos Elemer, Aug 12 2002

Extensions

Edited by Jon E. Schoenfield, Dec 10 2018

a(15)-a(18) from Amiram Eldar, Dec 11 2018

a(19)-a(33) from Chai Wah Wu, Jan 27 2020

Status

approved