A033844 -id:A033844 - cp's OEIS frontend

A060208 a(n) = 2pi(n) - pi(2n), where pi(i) = A000720(i).

-1, 0, 1, 0, 2, 1, 2, 2, 1, 0, 2, 1, 3, 3, 2, 1, 3, 3, 4, 4, 3, 2, 4, 3, 3, 3, 2, 2, 4, 3, 4, 4, 4, 3, 3, 2, 3, 3, 3, 2, 4, 3, 5, 5, 4, 4, 6, 6, 5, 5, 4, 3, 5, 4, 3, 3, 2, 2, 4, 4, 6, 6, 6, 5, 5, 4, 6, 6, 5, 4, 6, 6, 8, 8, 7, 6, 6, 6, 7, 7, 7, 6, 8, 7, 7, 7, 6, 6, 8, 7, 6, 6, 6, 6, 6, 5, 6, 6, 5, 4, 6, 6, 8, 8, 8, 7, 9, 9, 11, 11, 11, 10, 12, 11, 10, 10, 9, 9, 9, 8, 7, 7

Offset: 1

Labos Elemer, Mar 19 2001

sign

Rosser & Schoenfeld show 2*pi(x) > pi(2*x) for x > 10. - N. J. A. Sloane, Jul 03 2013, corrected Jul 09 2015

			n=100, pi(100)=25, pi(200)=46, 2pi(100)-pi(2*100) =4=a(100)

J. Barkley Rosser and Lowell Schoenfeld, Abstracts of Scientific Communications, Internat. Congress Math., Moscow, 1966, Section 3, Theory of Numbers.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.
Sanford Segal, On Pi(x+y)<=Pi(x)+Pi(y). Transactions American Mathematical Society, 104 (1962), 523-527.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271-274. Shows a(n)>0 for n>10.
E. Labos, Illustration

Cf. A000720, A007097, A033844, A060207, A212210, A212211, A212212, A212213.

[2*#PrimesUpTo(n) -#PrimesUpTo(2*n): n in [1..200]]; // G. C. Greubel, Aug 01 2024

f[n_] := 2 PrimePi[n] - PrimePi[2 n]; Array[f, 122] (* Robert G. Wilson v, Aug 12 2011 *)

a(n)=2*primepi(n)-primepi(2*n) \\ Charles R Greathouse IV, Jul 02 2013

[2*prime_pi(n) -prime_pi(2*n) for n in range(1,201)] # G. C. Greubel, Aug 01 2024

a(n) = Mod[2*PrimePi[n], PrimePi[2n]] = 2*A000720(n) - A000720(2n) for n>1.
a(n) ~ 2n log 2 / (log n)^2, by the prime number theorem. - N. J. A. Sloane, Mar 12 2007
a(n) = -A047886(n,n) (see A212210 to A212213). - Reinhard Zumkeller, Apr 15 2008

Edited by N. J. A. Sloane, Jul 03 2013

A099825 Sum of the first 2^n primes.

Original entry on oeis.org

2, 5, 17, 77, 381, 1851, 8893, 41741, 191755, 868151, 3875933, 17120309, 74950547, 325590115, 1405167561, 6029676711, 25750781177, 109495928099, 463852117169, 1958476902435, 8244703036797, 34615624751259, 144991244981985, 605994279458465, 2527803622205465

Offset: 0

Robert G. Wilson v, Oct 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..65 (calculated using Kim Walisch's primesum program; terms 0..30 from Robert G. Wilson v)
Cino Hilliard, Sumprimes. [broken link]
Kim Walisch, Sum of the primes below x (primesum).

Cf. A000040, A000079, A006988, A007504, A033844, A099824, A099826, A121248.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[0] = 2; f[n_] := f[n] = Block[{k = 0, mx = 2^n/2, np = Prime[2^n/2], s = f[n - 1]}, While[k < mx, k++; np = NextPrim@np; s = s + np]; s]; Table[ f@n, {n, 0, 23}] (* Robert G. Wilson v, Aug 24 2006 *)
Module[{nn=22,ap},ap=Accumulate[Prime[Range[2^nn]]];Table[ap[[2^n]],{n,0,nn}]] (* Harvey P. Dale, Apr 12 2017 *)

a(n)=my(s); n=2^n; forprime(p=2,, s+=p; if(n--==0, return(s))) \\ Charles R Greathouse IV, Feb 16 2017 \\ corrected by David A. Corneth, Aug 05 2025

a(n) = A007504(A000079(n)). - Amiram Eldar, Jul 01 2024

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

A062439 Primes at large indices: a(n) = prime(n!).

Original entry on oeis.org

2, 2, 3, 13, 89, 659, 5443, 49033, 484037, 5222429, 61194647, 774825383, 10552185239, 153903050137, 2394322471421, 39588599419319, 693389445083107, 12826386978604427, 249902442548157673, 5115640857307591139, 109776797549312197217, 2464348772728229970857

Offset: 0

Labos Elemer, Jul 09 2001

Andrew Booker, Nth Prime Page.

Cf. A000040, A000142, A033844.

[(NthPrime(Factorial(n))): n in [0..11]]; // Vincenzo Librandi, Dec 07 2018

seq(ithprime(factorial(n)),n=0..10); # Muniru A Asiru, Dec 07 2018

Array[Prime[#!] &, 16, 0] (* Michael De Vlieger, Dec 06 2018 *)

PARI
```
a(n) = prime(n!)
```

from sympy import prime, factorial
for n in range(0,14): print(prime(factorial(n))) # Stefano Spezia, Dec 07 2018

a(n) = A000040(A000142(n)).

a(15)-a(19) from Jens Kruse Andersen, May 08 2010

a(20)-a(21) from Henri Lifchitz, Sep 09 2014

A325094 Write n as a sum of distinct powers of 2, then take the primes of those powers of 2 and multiply them together.

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 21, 42, 19, 38, 57, 114, 133, 266, 399, 798, 53, 106, 159, 318, 371, 742, 1113, 2226, 1007, 2014, 3021, 6042, 7049, 14098, 21147, 42294, 131, 262, 393, 786, 917, 1834, 2751, 5502, 2489, 4978, 7467, 14934, 17423, 34846, 52269, 104538, 6943

Offset: 0

Gus Wiseman, Mar 27 2019

The sorted sequence is A325093.

For example, 11 = 1 + 2 + 8, so a(11) = prime(1) * prime(2) * prime(8) = 114.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    6: {1,2}
    7: {4}
   14: {1,4}
   21: {2,4}
   42: {1,2,4}
   19: {8}
   38: {1,8}
   57: {2,8}
  114: {1,2,8}
  133: {4,8}
  266: {1,4,8}
  399: {2,4,8}
  798: {1,2,4,8}
   53: {16}
  106: {1,16}
  159: {2,16}
  318: {1,2,16}
  371: {4,16}

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000720, A001222, A005117, A018819, A019565, A033844, A056239, A102378, A112798, A247935, A318400.

Cf. A325091, A325092, A325093, A325106.

P:= [seq(ithprime(2^i),i=0..10)]:
f:= proc(n) local L,i;
  L:= convert(n,base,2);
  mul(P[i]^L[i],i=1..nops(L))
end proc:
map(f, [$0..100]); # Robert Israel, Mar 28 2019

Table[Times@@MapIndexed[If[#1==0,1,Prime[2^(#2[[1]]-1)]]&,Reverse[IntegerDigits[n,2]]],{n,0,100}]

A119772 a(n) = prime(4^n).

Original entry on oeis.org

2, 7, 53, 311, 1619, 8161, 38873, 180503, 821641, 3681131, 16290047, 71378569, 310248241, 1339484197, 5750079047, 24563311309, 104484802057, 442795487221, 1870358526653, 7877263558621, 33089240375501, 138666449011757, 579863159340527, 2420094683001859, 10082409897709157

Offset: 0

Jim Snow (jsnow(AT)mitre.org), Jun 22 2006

Amiram Eldar, Table of n, a(n) for n = 0..39 (calculated from the b-file at A033844; terms 0..28 from Charles R Greathouse IV, a(26) corrected)
Andrew Booker, The Nth Prime Page.

Cf. A033844.

Prime[4^Range[0,20]] (* Harvey P. Dale, Jul 12 2011 *)

a(n)=prime(4^n) \\ Charles R Greathouse IV, Nov 02 2014

a(n) = A033844(2*n). - Amiram Eldar, Jun 06 2024

a(20)-a(24) from Charles R Greathouse IV, Nov 02 2014

A324927 Matula-Goebel numbers of rooted trees of depth 2. Numbers that are not powers of 2 but whose prime indices are all powers of 2.

Original entry on oeis.org

3, 6, 7, 9, 12, 14, 18, 19, 21, 24, 27, 28, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 72, 76, 81, 84, 96, 98, 106, 108, 112, 114, 126, 131, 133, 144, 147, 152, 159, 162, 168, 171, 189, 192, 196, 212, 216, 224, 228, 243, 252, 262, 266, 288, 294, 304, 311, 318

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Numbers n such that A109082(n) = 2.
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.
Also Heinz numbers of integer partitions into powers of 2 with at least one part > 1 (counted by A102378).

			The sequence of terms together with their prime indices begins:
   3: {2}
   6: {1,2}
   7: {4}
   9: {2,2}
  12: {1,1,2}
  14: {1,4}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  24: {1,1,1,2}
  27: {2,2,2}
  28: {1,1,4}
  36: {1,1,2,2}
  38: {1,8}
  42: {1,2,4}
  48: {1,1,1,1,2}
  49: {4,4}
  53: {16}
  54: {1,2,2,2}
  56: {1,1,1,4}

Cf. A000081, A000720, A003963, A007097, A018819, A033844, A056239, A102378, A112798, A302242, A318400, A324928, A324929.

Select[Range[100],And[!IntegerQ[Log[2,#]],And@@Cases[FactorInteger[#],{p_,_}:>IntegerQ[Log[2,PrimePi[p]]]]]&]

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

A336321 a(n) = A122111(A225546(n)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 19, 6, 9, 11, 53, 10, 131, 23, 13, 8, 311, 15, 719, 22, 29, 59, 1619, 14, 49, 137, 21, 46, 3671, 17, 8161, 12, 61, 313, 37, 25, 17863, 727, 139, 26, 38873, 31, 84017, 118, 39, 1621, 180503, 20, 361, 77, 317, 274, 386093, 33, 71, 58, 733, 3673, 821641, 34, 1742537, 8167, 87, 18, 151, 67, 3681131, 626, 1627, 41, 7754077, 35, 16290047

Offset: 1

Antti Karttunen and Peter Munn, Jul 17 2020

nonn

A122111 and A225546 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A122111 maps the k-th prime to 2^k, whereas A225546 maps it to 2^2^(k-1).

In composing these permutations, this sequence maps the squarefree numbers, as listed in A019565, to the prime numbers in increasing order; and the list of powers of 2 to the "normal" numbers (A055932), as listed in A057335.

			From _Peter Munn_, Jan 04 2021: (Start)
In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
First, a table showing mapping of the powers of 2:
  n     a^-1(2^n) =    2^n =        a(2^n) =
        A001146(n-1)   A000079(n)   A057335(n)
  0             (1)         1            1
  1               2         2            2
  2               4         4            4
  3              16         8            6
  4             256        16            8
  5           65536        32           12
  6      4294967296        64           18
  ...
Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
  n   a^-1(A019565(n))   A019565(n)      a(A019565(n))   a^2(A019565(n))
      Cf. {A337533}      Cf. {A005117}   = prime(n)      = A033844(n-1)
  0              1               1             (1)               (1)
  1              2               2               2                 2
  2              3               3               3                 3
  3              8               6               5                 7
  4              6               5               7                19
  5             12              10              11                53
  6             18              15              13               131
  7            128              30              17               311
  8              5               7              19               719
  9             24              14              23              1619
  ...
As sets, the above columns are A337533, A005117, A008578, {1} U A033844.
Similarly, we get bijections between sets A000290\{0} -> {1} U A070003; and {1} U A335740 -> A005408 -> A066207.
(End)

A122111 composed with A225546.
Cf. A336322 (inverse permutation).
Other sequences used in a definition of this sequence: A000040, A000188, A019565, A248663, A253550, A253560.
Sequences used to express relationship between terms of this sequence: A003159, A003961, A297002, A334747.
Cf. A057335.
A mapping between the binary tree sequences A334866 and A253563.
Lists of sets (S_1, S_2, ... S_j) related by the bijection defined by the sequence: (A000290\{0}, {1} U A070003), ({1} U A001146, A000079, A055932), ({1} U A335740, A005408, A066207), (A337533, A005117, A008578, {1} U A033844).

a(n) = A122111(A225546(n)).
Alternative definition: (Start)
Write n = m^2 * A019565(j), where m = A000188(n), j = A248663(n).
a(1) = 1; otherwise for m = 1, a(n) = A000040(j), for m > 1, a(n) = A253550^j(A253560(a(m))).
(End)
a(A000040(m)) = A033844(m-1).
a(A001146(m)) = 2^(m+1).
a(2^n) = A057335(n).
a(n^2) = A253560(a(n)).
For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
More generally, a(A334747(n)) = b(a(n)).
a(A003961(n)) = A297002(a(n)).
a(A334866(m)) = A253563(m).

cp's OEIS Frontend

A060208 a(n) = 2*pi(n) - pi(2*n), where pi(i) = A000720(i).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A099825 Sum of the first 2^n primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A062439 Primes at large indices: a(n) = prime(n!).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A325094 Write n as a sum of distinct powers of 2, then take the primes of those powers of 2 and multiply them together.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A119772 a(n) = prime(4^n).

Views

Author

Keywords

Links

Crossrefs

A060208 a(n) = 2pi(n) - pi(2n), where pi(i) = A000720(i).