A051438 -id:A051438 - 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

A051439 a(n) = prime(2^n + 1).

Original entry on oeis.org

3, 5, 11, 23, 59, 137, 313, 727, 1621, 3673, 8167, 17881, 38891, 84047, 180511, 386117, 821647, 1742539, 3681149, 7754081, 16290073, 34136059, 71378603, 148948141, 310248251, 645155227, 1339484207, 2777105131, 5750079077

Offset: 0

N. J. A. Sloane

nonn

			a(0) = prime(2^0+1) = prime(2) = 3,
a(1) = prime(2+1) = prime(3) = 5,
a(2) = prime(2^2+1) = prime(5) = 11,
a(3) = prime(2^3+1) = prime(9) = 23, and so on. - _N. J. A. Sloane_, Dec 09 2020

Hugo Pfoertner, Table of n, a(n) for n = 0..78, using David Baugh's b-file from A033844.
Kim Walisch, Fast C++ prime counting function implementation.

Cf. A000040, A033844, A018249, A051438, A051440.

Prime[2^Range[0,30]+1] (* Harvey P. Dale, Jan 19 2014 *)

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

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

A181363 1 followed by the primes, interleaved recursively.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 7, 3, 11, 2, 13, 5, 17, 1, 19, 7, 23, 3, 29, 11, 31, 2, 37, 13, 41, 5, 43, 17, 47, 1, 53, 19, 59, 7, 61, 23, 67, 3, 71, 29, 73, 11, 79, 31, 83, 2, 89, 37, 97, 13, 101, 41, 103, 5, 107, 43, 109, 17, 113, 47, 127, 1, 131, 53, 137, 19, 139, 59, 149, 7, 151, 61

Offset: 1

Reinhard Zumkeller, Oct 16 2010

nonn

a(2*n-1) = A008578(n); a(2*n+1) = A000040(n);
a((2*n-1)*2^k) = A008578(n), k >= 0;
a(2^k) = 1, k >= 0; a(2^k - 1) = A051438(k), k > 0.

			Initial values, seen as a binary tree:
................................. 1
................ 1 _______________________________ 2
........ 1 _____________ 3 .............. 2 ______________ 5
... 1 _____ 7 ..... 3 ______ 11 .... 2 ______ 13 .... 5 ______ 17
.. 1__19...7__23...3__29...11__31...2__37...13__41...5__43...17__47

R. Zumkeller, Table of n, a(n) for n = 1..8191

import Data.List (transpose)
a181363 n = a181363_list !! (n-1)
a181363_list = concat $ transpose [a008578_list, a181363_list]
-- Reinhard Zumkeller, Mar 22 2014, Oct 20 2011, Oct 16 2010

a(n) = if even n then a(n/2) else A008578((n+1)/2).

a(n) = A008578(A025480(n-1)+1). - Reinhard Zumkeller, Mar 22 2014

A325122 Sum of binary digits of the prime indices of n, minus Omega(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 1, 2, 0, 1, 2, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 3, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 2, 0, 0, 2, 1, 2, 2, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 2, 1, 0, 2, 3, 0, 3, 2, 1

Offset: 1

Gus Wiseman, Mar 29 2019

nonn

The sum of binary digits of an integer is the number of 1's in its binary representation. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Positions of zeros are A318400.
Cf. A000120, A018900, A019565, A048881, A051438, A070939, A112798.
Cf. A325103, A325104, A325106, A325118.
Other totally additive sequences: A056239, A302242, A318994, A318995, A325033, A325034, A325120, A325121.

Table[Sum[pr[[2]]*(DigitCount[PrimePi[pr[[1]]],2,1]-1),{pr,If[n==1,{},FactorInteger[n]]}],{n,100}]

Totally additive with a(prime(n)) = A048881(n).

A050298 Triangle read by rows: T(n,k) = p(r), where r is the (n-k+1)-th number such that A001222(r+1) = k, and p(r) is the r-th prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 13, 19, 31, 47, 29, 23, 59, 83, 127, 37, 41, 67, 149, 211, 307, 53, 43, 101, 167, 353, 499, 709, 61, 71, 103, 241, 401, 823, 1153, 1613, 79, 73, 109, 257, 587, 937, 1873, 2647, 3659, 107, 89, 179, 277, 607, 1319, 2113, 4201, 5843, 8147

Offset: 1

Alford Arnold, Apr 09 2003

The first column is A055003 and the main diagonal is A051438. When viewed as a sequence, this is a permutation of the prime numbers.

			a(14) = T(5,4) = p(23) = 83 because A001222(23+1) = A001222(24) = 4 since 24 has four prime factors, and this is the (5-4+1) = 2nd number with A001222 = 4.
The table begins:
2
3  5
7  11 17
13 19 31 47
29 23 59 83  127
37 41 67 149 211 307
...

Nathaniel Johnston, Rows n=1..20, flattened

Cf. A000040, A001222, A064553.

with(numtheory): A050298ind := proc(n,k) option remember: local f,m: if(n=k)then return 2^n-1: fi: for m from procname(n-1,k)+1 do if(bigomega(m+1)=k)then return m: fi: od: end: for n from 1 to 6 do seq(ithprime(A050298ind(n,k)),k=1..n);od; # Nathaniel Johnston, May 07 2011

Better name and extended by Nathaniel Johnston, May 07 2011

cp's OEIS Frontend

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

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A051439 a(n) = prime(2^n + 1).

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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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A181363 1 followed by the primes, interleaved recursively.

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A325122 Sum of binary digits of the prime indices of n, minus Omega(n).

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A050298 Triangle read by rows: T(n,k) = p(r), where r is the (n-k+1)-th number such that A001222(r+1) = k, and p(r) is the r-th prime.

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