A051439 -id:A051439 - cp's OEIS frontend

A033844 a(n) = prime(2^n).

2, 3, 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: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

a(n) is the smallest number m such that pi(m)=d(m)^n, where d(m) is number of positive divisors of m (the proof is easy). - Farideh Firoozbakht, Jun 06 2005

David Baugh, Table of n, a(n) for n = 0..78 (terms n = 0..57 from Charles R Greathouse IV, terms n = 58..78 found using Kim Walisch's primecount program).

Cf. A000040, A018249, A051438, A051440, A051439, A006988, A107655.

Mathematica
```
Table[Prime[2^n], {n, 0, 32}]
```

a(n)=prime(2^n) \\ Charles R Greathouse IV, Nov 02 2014

More terms from Robert G. Wilson v, Jun 09 2000

A082862 Prime(2^j) where j are the positions at which (prime(2^k+1)-prime(2^k))/log(prime(2^k)) set low-value records.

Original entry on oeis.org

2, 19, 131, 311, 1619, 3671, 1742537, 148948139, 2777105129, 16149760533341, 10082409897709157

Offset: 1

Labos Elemer, Apr 14 2003

Similar to but not identical to A074327.

			The values of quotients at primes of this sequence are as follows: 1.442695..., 1.358493..., 1.230719..., 0.348444..., 0.270651..., 0.243658..., 0.139170..., 0.106274..., 0.091976..., 0.065761..., 0.054274... .

American Institute of Mathematics, San Jose State Math Researcher Experiences Epiphany at American Institute Of Mathematics, March 21, 2003. [Wayback Machine link]
American Institute of Mathematics, Small gaps between consecutive primes. [Wayback Machine link]
American Institute of Mathematics, On the error in Goldston and Yildirim's "Small gaps between consecutive primes". [Wayback Machine link]

Cf. A033844, A051439, A074327.

q=4; Do[s=(Prime[2^n+1]-Prime[2^n])/Log[Prime[2^n]]//N; If[sAmiram Eldar, Aug 10 2024 *)

Data corrected and extended by Amiram Eldar, Aug 10 2024

A051438 a(n) = prime(2^n - 1).

Original entry on oeis.org

2, 5, 17, 47, 127, 307, 709, 1613, 3659, 8147, 17851, 38867, 84011, 180497, 386083, 821603, 1742527, 3681113, 7754017, 16290041, 34136021, 71378551, 148948133, 310248233, 645155191, 1339484149, 2777105117, 5750079043, 11891268397, 24563311217, 50685770143

Offset: 1

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 1..78 (calculated using the b-file at A033844; terms 1..57 from Hugo Pfoertner)

Cf. A000040, A000225, A033844, A018249, A051440, A051439, A151799.

Prime[2^Range[34] - 1] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

a(n)=prime(2^n-1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = A181363(2^n - 1). - Reinhard Zumkeller, Oct 16 2010
a(n) = A000040(A000225(n)). - Michel Marcus, Nov 28 2017
a(n) = A151799(A033844(n)). - Amiram Eldar, Jun 30 2024

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

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

Original entry on oeis.org

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

A018249 a(n) = prime(2^n)-1.

Original entry on oeis.org

1, 2, 6, 18, 52, 130, 310, 718, 1618, 3670, 8160, 17862, 38872, 84016, 180502, 386092, 821640, 1742536, 3681130, 7754076, 16290046, 34136028, 71378568, 148948138, 310248240, 645155196, 1339484196, 2777105128, 5750079046, 11891268400, 24563311308, 50685770166, 104484802056

Offset: 0

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 0..78
Paul Cooijmans, Numbers, Item 18, 1995.
Paul Cooijmans, Short Test For Genius, Item 29, 2002.

Cf. A000040, A033844, A051438, A051440, A051439.

Prime[2^Range[0,30]]-1 (* Harvey P. Dale, Sep 30 2016 *)

a(n) = prime(2^n) - 1; \\ Amiram Eldar, Jul 11 2025

a(n) = A033844(n) - 1.

a(n) = A051440(n) - 2. - Amiram Eldar, Jul 11 2025

More terms from Amiram Eldar, Jul 11 2025

A051440 a(n) = prime(2^n) + 1.

Original entry on oeis.org

3, 4, 8, 20, 54, 132, 312, 720, 1620, 3672, 8162, 17864, 38874, 84018, 180504, 386094, 821642, 1742538, 3681132, 7754078, 16290048, 34136030, 71378570, 148948140, 310248242, 645155198, 1339484198, 2777105130, 5750079048, 11891268402, 24563311310, 50685770168, 104484802058

Offset: 0

N. J. A. Sloane

nonn

Amiram Eldar, Table of n, a(n) for n = 0..78

Cf. A000040, A018249, A033844, A051438, A051439.

a[n_] := Prime[2^n] + 1; Array[a, 33, 0] (* Amiram Eldar, Jul 11 2025 *)

a(n) = prime(2^n) + 1; \\ Amiram Eldar, Jul 11 2025

a(n) = A033844(n) + 1.

a(n) = A018249(n) + 2. - Amiram Eldar, Jul 11 2025

More terms from Amiram Eldar, Jul 11 2025

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

Original entry on oeis.org

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

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

A033844 a(n) = prime(2^n).

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A082862 Prime(2^j) where j are the positions at which (prime(2^k+1)-prime(2^k))/log(prime(2^k)) set low-value records.

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A051438 a(n) = prime(2^n - 1).

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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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A018249 a(n) = prime(2^n)-1.

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A051440 a(n) = prime(2^n) + 1.

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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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