A073798 -id:A073798 - cp's OEIS frontend

A074325 Difference between (1+2^n)-th and (2^n)-th primes. Also number of terms in blocks of A073798.

1, 2, 4, 4, 6, 6, 2, 8, 2, 2, 6, 18, 18, 30, 8, 24, 6, 2, 18, 4, 26, 30, 34, 2, 10, 30, 10, 2, 30, 6, 70, 4, 12, 22, 6, 24, 26, 10, 88, 2, 50, 18, 6, 20, 14, 4, 12, 6, 2, 56, 4, 30, 42, 6, 70, 74, 14, 60, 170, 14, 44, 22, 52, 36, 96, 6, 86, 86, 72, 48, 42, 18, 20, 24, 10, 154, 54, 20, 12

Offset: 0

Labos Elemer, Aug 21 2002

			For n = 39: 2^39 = 549755813888, prime(549755813889) = 16149760533343, prime(549755813888) =  16149760533341, difference = 2 (just twin primes), so a(39) = 2.

Cf. A051439, A033844, A073798, A000079, A001223.

Table[Prime[1+2^n] - Prime[2^n], {n, 1, 39}]

a(n) = A051439(n) - A033844(n).

a(n) = A001223(A000079(n)). - Michel Marcus, May 10 2024

More terms from Michel ten Voorde Jun 13 2003
a(42) from Max Alekseyev, May 08 2009
Offset corrected by Max Alekseyev, Oct 24 2013
a(43)-a(57) from Chai Wah Wu, Aug 30 2019
a(58)-a(78) calculated using the data at A033844 and A051439 and added by Amiram Eldar, May 10 2024

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 *)

A074326 Numbers n such that difference between (1+2^n)-th and (2^n)-th primes is 2.

Original entry on oeis.org

1, 6, 8, 9, 17, 23, 27, 39, 48

Offset: 1

Labos Elemer, Aug 21 2002

			n=39: 2^39=549755813888, prime(549755813889) = 16149760533343, prime(549755813888) = 16149760533341, difference=2, just twin primes.

Cf. A029707, A051439, A033844, A073798, A000079.

s=0; Do[s=Prime[1+2^n]-Prime[2^n]; If[s==2, Print[{n, Prime[2^n]}]], {n, 1, 40}]
diffQ[n_]:=Module[{prn=Prime[2^n]},NextPrime[prn]-prn==2]; Select[ Range[ 40],diffQ] (* Harvey P. Dale, Aug 21 2014 *)

Solutions to A051439(x) - A033844(x) = 2,

a(9) from Chai Wah Wu, Feb 28 2019

A074327 Smaller of twin primes arising in A074326.

Original entry on oeis.org

3, 311, 1619, 3671, 1742537, 148948139, 2777105129, 16149760533341, 10082409897709157

Offset: 1

Labos Elemer, Aug 21 2002

			8th term in A074326 is 39, so prime[549755813888]=16149760533341=a(8) and 16149760533343 is the larger twin.

Cf. A051439, A033844, A073798, A000079, A074325, A074326.

s=0; Do[s=Prime[1+2^n]-Prime[2^n]; If[s==2, Print[Prime[2^n]]], {n, 1, 40}]

a(n) = prime(m) where m=2^x and prime(m+1)-prime(m) = 2.

a(n) = A033844(A074326(n)). - Michel Marcus, Aug 28 2019

Corrected and extended by Michel Marcus, Aug 28 2019

cp's OEIS Frontend

A074325 Difference between (1+2^n)-th and (2^n)-th primes. Also number of terms in blocks of A073798.

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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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A074326 Numbers n such that difference between (1+2^n)-th and (2^n)-th primes is 2.

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A074327 Smaller of twin primes arising in A074326.

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