A034785 -id:A034785 - cp's OEIS frontend

A061285 a(n) = 2^((prime(n) - 1)/2).

2, 4, 8, 32, 64, 256, 512, 2048, 16384, 32768, 262144, 1048576, 2097152, 8388608, 67108864, 536870912, 1073741824, 8589934592, 34359738368, 68719476736, 549755813888, 2199023255552, 17592186044416, 281474976710656, 1125899906842624, 2251799813685248

Offset: 2

Labos Elemer, May 22 2001

nonn

Square root of 2^(prime(n) - 1), i.e., the smallest number that has prime(n) divisors.

Index to divisibility sequences

Cf. A000005, A000010, A005097, A005179, A006005, A006093, A034785, A217054.

Table[2^((Prime[n] - 1)/2), {n, 2, 25}] (* Amiram Eldar, Dec 23 2020 *)

a(n) = sqrt(min(x; A000005(x) = prime(n))) = sqrt(A034785(n)/2) = sqrt(2^(prime(n) - 1)) = sqrt(2^A006093(n)) = sqrt(2^phi(prime(n))) = sqrt(2^A000010(A000040(n))).

Sum_{n>=1} 1/a(n) = A217054. - Amiram Eldar, Dec 23 2020

A098096 Numbers of the form p^2 * 2^p for p prime.

Original entry on oeis.org

16, 72, 800, 6272, 247808, 1384448, 37879808, 189267968, 4437573632, 451508436992, 2063731785728, 188153927303168, 3696558092582912, 16263975998062592, 310889111776919552, 25301222706567446528

Offset: 1

Parthasarathy Nambi, Sep 14 2004

a(n) = A001248(n) * A034785(n). - Reinhard Zumkeller, Feb 07 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..450

Cf. A082949, numbers of the form p^q * q^p, with distinct primes p and q.

Cf. A001248, A034785.

a098096 n = a001248 n * a034785 n  -- Reinhard Zumkeller, Feb 07 2015

Table[2^Prime[p]*Prime[p]^2, {p, 16}] (* Alonso del Arte, Oct 28 2005 *)

PARI
```
forprime(p=2,53,print1(p^2*2^p,","))
```

More terms from Klaus Brockhaus and Ray Chandler, Sep 15 2004

A100042 a(n) = prime(n)*2^prime(n).

Original entry on oeis.org

8, 24, 160, 896, 22528, 106496, 2228224, 9961472, 192937984, 15569256448, 66571993088, 5085241278464, 90159953477632, 378231999954944, 6614661952700416, 477381560501272576, 34011184385901985792

Offset: 1

Jorge Coveiro, Dec 26 2004

nonn

A lower bound for the 2-color van der Waerden number A005346(prime(n)); see Berlekamp reference. - Charles R Greathouse IV, Jul 13 2008

E. R. Berlekamp, A construction for partitions which avoid long arithmetic progressions, Canadian Mathematical Bulletin, Vol. 11, No. 3 (1968), pp. 409-414.

Cf. A005346, A034785, A157413.

Maple
```
seq(ithprime(n)*2^ithprime(n),n=1..20);
```

#*2^#&/@Prime[Range[20]] (* Harvey P. Dale, Sep 18 2020 *)

a(n) = prime(n)*2^prime(n); \\ Michel Marcus, May 14 2017

Sum_{n>=1} 1/a(n) = A157413. - Amiram Eldar, Nov 17 2020

A100105 a(n) = 2^prime(n)-prime(n).

Original entry on oeis.org

2, 5, 27, 121, 2037, 8179, 131055, 524269, 8388585, 536870883, 2147483617, 137438953435, 2199023255511, 8796093022165, 140737488355281, 9007199254740939, 576460752303423429, 2305843009213693891

Offset: 1

Jorge Coveiro, Dec 26 2004

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..100

Cf. A001348, A034785.

[2^p - p: p in PrimesUpTo(80)]; // Vincenzo Librandi, Jun 03 2014

seq( (2^ithprime(x)-ithprime(x)),x=1..20);

f[n_]:=(2^Prime[n]-Prime[n]); Array[f,4!,1] (* Vladimir Joseph Stephan Orlovsky, Oct 17 2009 *)

A135171 3^p - 2^p, where p = prime(n).

Original entry on oeis.org

5, 19, 211, 2059, 175099, 1586131, 129009091, 1161737179, 94134790219, 68629840493971, 617671248800299, 450283768452043891, 36472994178147530851, 328256958598444055419, 26588814218220014932459, 19383245658672820642055731, 14130386091162273752461387579, 127173474823342767533669605651

Offset: 1

Omar E. Pol, Nov 25 2007

nonn

			a(4)=2059 because the 4th prime number is 7, 3^7=2187, 2^7=128 and 2187-128=2059.

Ivan Panchenko, Table of n, a(n) for n = 1..200

Cf. A000040, A034785, A057901.

[3^p-2^p: p in PrimesUpTo(100)]; // Vincenzo Librandi, Dec 14 2010

3^#-2^#&/@Prime[Range[20]] (* Harvey P. Dale, Jun 20 2011 *)

a(n) = 3^A000040(n) - 2^A000040(n).

a(n) = A057901(n) - A034785(n). - Michel Marcus, Jun 14 2014

More terms from Vincenzo Librandi, Dec 14 2010

A152099 a(n) = (2^prime(n) - 1)(2^prime(n) + 1) = 2^(2prime(n)) - 1.

Original entry on oeis.org

15, 63, 1023, 16383, 4194303, 67108863, 17179869183, 274877906943, 70368744177663, 288230376151711743, 4611686018427387903, 18889465931478580854783, 4835703278458516698824703, 77371252455336267181195263, 19807040628566084398385987583

Offset: 1

Roger L. Bagula, Nov 24 2008

nonn

Idea resulted from seqfan posts by Artur Jasinski.

Cf. A000040, A000302, A001348, A034785, A098640.

Table[(2^Prime[n] - 1)*(2^Prime[n] + 1), {n, 1, 20}]

from sympy import prime
def A152099(n): return (1<<(prime(n)<<1))-1 # Chai Wah Wu, Jun 26 2023

a(n) = A001348(n) * A098640(n).
a(n) = A034785(n)^2 - 1.
a(n) = A000302(A000040(n)) - 1.

A086690 Number of primes < 2^prime(n).

Original entry on oeis.org

2, 4, 11, 31, 309, 1028, 12251, 43390, 564163, 28192750, 105097565, 5586502348, 80316571436, 305761713237, 4461632979717, 252252704148404, 14458792895301660, 55890484045084135, 3249254387052557215, 48995571600129458363, 190499823401327905601

Offset: 1

Cino Hilliard, Jul 28 2003

nonn

			There are 11 primes < 2^prime(3) = 2^5, so a(3) = 11.

Amiram Eldar, Table of n, a(n) for n = 1..24 (terms 1..23 from Gord Palameta)
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x).

Cf. A000040, A000720, A007053, A034785.

Table[ PrimePi[ 2^Prime[n]], {n, 1, 14}]

PARI
```
a(n) = primepi(2^prime(n));
```

a(n) = A007053(A000040(n)).

a(n) = A000720(A034785(n)). - Amiram Eldar, Jul 06 2024

Extended by Robert G. Wilson v, Jul 30 2003

Revised by Max Alekseyev, Jul 20 2007

A121240 Numerator of sum_{k=1..n} 1/2^prime(k).

Original entry on oeis.org

1, 3, 13, 53, 849, 3397, 54353, 217413, 3478609, 222630977, 890523909, 56993530177, 911896482833, 3647585931333, 58361374901329, 3735127993685057, 239048191595843649, 956192766383374597, 61196337048535974209

Offset: 1

Alexander Adamchuk, Aug 22 2006

a(n) is prime for n = {2, 3, 4, 10, 21, 321,..} where it takes the values {3, 13, 53, 222630977, ...}.

The prime constant A051006 = 0.414682509.. is limit(n->infinity) a(n)/2^prime(n) .

Eric Weisstein's World of Mathematics, Prime Constant.

Cf. A034785 (denominators), A072762, A051006, A010051.

Table[Numerator[Sum[1/2^Prime[k],{k,1,n}]],{n,1,30}]
Accumulate[1/2^Prime[Range[30]]]//Numerator (* Harvey P. Dale, Aug 11 2021 *)

a(n) = Numerator[ Sum[ 1/2^Prime[k], {k,1,n} ] ]. a(n) = A072762[ Prime[n] ].

A135175 a(n) = 5^p + 3^p - 2^p, where p = prime(n).

Original entry on oeis.org

30, 144, 3336, 80184, 49003224, 1222289256, 763068462216, 19074648065304, 11921023089868344, 186264583552936197096, 4656613490748641378424, 72759576592118027485247016, 45474735125119406073899483976, 1136868377544417255992242883544, 710542735786689000089344282510584

Offset: 1

Omar E. Pol, Nov 25 2007

			a(4)=80184 because the 4th prime number is 7, 5^7=78125, 3^7=2187, 2^7=128 and 78125+2187-128=80184.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. 2^p: A034785. 3^p: A057901. 2^5: A057902.

[5^p+3^p-2^p: p in PrimesUpTo(100)]; // Vincenzo Librandi, Dec 14 2010

a:= n-> (p-> 5^p+3^p-2^p)(ithprime(n)):
seq(a(n), n=1..15);  # Alois P. Heinz, Jun 08 2025

5^#+3^#-2^#&/@Prime[Range[20]]  (* Harvey P. Dale, Apr 04 2011 *)
Table[5^p + 3^p - 2^p, {p, Prime[Range[20]]}] (* Vincenzo Librandi, May 24 2014 *)

a(n) = 5^p + 3^p - 2^p with p = A000040(n).

More terms from Vincenzo Librandi, Dec 14 2010

A135620 a(n) = 2^(prime(n) - 2).

Original entry on oeis.org

1, 2, 8, 32, 512, 2048, 32768, 131072, 2097152, 134217728, 536870912, 34359738368, 549755813888, 2199023255552, 35184372088832, 2251799813685248, 144115188075855872, 576460752303423488, 36893488147419103232, 590295810358705651712, 2361183241434822606848, 151115727451828646838272

Offset: 1

Omar E. Pol, Mar 01 2008

G. C. Greubel, Table of n, a(n) for n = 1..465

Cf. A000040, A034785, A040976, A061286.

Partial differences of A135482.

[2^(NthPrime(n)-2): n in [1..25]]; // Vincenzo Librandi, Oct 23 2016

Table[2^(Prime[n] - 2), {n,1,25}] (* G. C. Greubel, Oct 23 2016 *)

a(n) = 2^(prime(n)-2); \\ Michel Marcus, Oct 23 2016

a(n) = 2^(A000040(n)-2) = 2^(A040976(n)) = 2^A000040(n)/4 = A061286(n)/2.

a(n) = A034785(n)/4. - Alois P. Heinz, Jun 08 2025

cp's OEIS Frontend

A061285 a(n) = 2^((prime(n) - 1)/2).

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A098096 Numbers of the form p^2 * 2^p for p prime.

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A100042 a(n) = prime(n)*2^prime(n).

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Maple

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A100105 a(n) = 2^prime(n)-prime(n).

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Magma

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A135171 3^p - 2^p, where p = prime(n).

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Magma

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Extensions

A152099 a(n) = (2^prime(n) - 1)*(2^prime(n) + 1) = 2^(2*prime(n)) - 1.

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Python

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A086690 Number of primes < 2^prime(n).

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Mathematica

PARI

Formula

Extensions

A121240 Numerator of sum_{k=1..n} 1/2^prime(k).

A152099 a(n) = (2^prime(n) - 1)(2^prime(n) + 1) = 2^(2prime(n)) - 1.