A073797 -id:A073797 - cp's OEIS frontend

A073798 pi(n) is a power of 2, where pi(n) = A000720(n) is the number of primes <= n.

2, 3, 4, 7, 8, 9, 10, 19, 20, 21, 22, 53, 54, 55, 56, 57, 58, 131, 132, 133, 134, 135, 136, 311, 312, 719, 720, 721, 722, 723, 724, 725, 726, 1619, 1620, 3671, 3672, 8161, 8162, 8163, 8164, 8165, 8166, 17863, 17864, 17865, 17866, 17867, 17868, 17869, 17870

Offset: 1

Labos Elemer, Aug 14 2002

nonn

The numbers occur in blocks of consecutive integers: 2, 3-4, 7-10, 19-22, ...; the n-th block starts at the 2^n-th prime (A033844) and ends just before the (2^n + 1)-th prime (A051439).

			10 is in the sequence since pi(10)=4=2^2.

Chai Wah Wu, Table of n, a(n) for n = 1..1103 (n = 1..665 from Ivan Neretin)

Cf. A000079, A000720, A015910, A073797, A073799.

pow2[n_] := n==1||(n>1&&IntegerQ[n/2]&&pow2[n/2]); Select[Range[20000], pow2[PrimePi[ # ]]&]
Flatten@Table[Range[p = Prime[2^k], NextPrime[p] - 1], {k, 0, 11}] (* Ivan Neretin, Jan 21 2017 *)

isok(n) = my(pi = primepi(n)); (pi==1) || (pi==2) || (ispower(primepi(n),,&k) && (k==2)); \\ Michel Marcus, Jan 23 2017

Edited by Dean Hickerson, Aug 15 2002

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

Original entry on oeis.org

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

Labos Elemer, Aug 12 2002

nonn

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)

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

Cf. A000079, A000720, A015910, A062173, A064367, A073797, A073798.
Cf. A033844. - R. J. Mathar, Sep 23 2008
Cf. A051439, A074325.

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 *)

a(n) = if(n==1, 2, prime(2^n)); \\ Jinyuan Wang, Mar 01 2020

from sympy import prime
def A073799(n):
    return 2 if n == 1 else prime(2**n) # Chai Wah Wu, Jan 27 2020

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

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

A073800 Remainder of division 2^n/c(n), where c(n)=A002808(n), the n-th composite.

Original entry on oeis.org

2, 4, 0, 7, 2, 4, 2, 1, 0, 16, 8, 1, 8, 16, 18, 16, 14, 8, 8, 0, 2, 30, 18, 28, 14, 4, 8, 16, 28, 19, 6, 16, 29, 34, 8, 40, 2, 14, 8, 16, 14, 4, 8, 4, 0, 49, 62, 52, 32, 4, 8, 46, 17, 20, 65, 22, 32, 16, 62, 64, 32, 64, 41, 16, 32, 64, 48, 70, 48, 24, 32, 22, 74, 84, 8, 16, 32, 52

Offset: 1

Labos Elemer, Aug 12 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A000079, A002808, A015910, A062173, A064367, A073797, A073798, A073799.

Table[Mod[2^j, FixedPoint[j+PrimePi[ # ]+1&, j]], {j, 1, 128}]
Module[{c=Select[Range[200],CompositeQ],len},len=Length[c];Table[ PowerMod[ 2,n,c[[n]]],{n,len}]] (* Harvey P. Dale, Mar 03 2018 *)

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A073798 pi(n) is a power of 2, where pi(n) = A000720(n) is the number of primes <= n.

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

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A073800 Remainder of division 2^n/c(n), where c(n)=A002808(n), the n-th composite.

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Mathematica