A034785 -id:A034785 - cp's OEIS frontend

A175613 Number of semiprimes <= 2^prime(n).

1, 2, 10, 42, 589, 2186, 30253, 113307, 1608668, 88157689, 336717854, 19015826478, 282528883551, 1091574618496, 16360940729894

Offset: 1

Juri-Stepan Gerasimov, Dec 04 2010

			a(2)=2 because first 2 semiprimes are 4, 6 both <2^prime(2)=8.

Cf. A001358, A007053, a proper subset of A125527.

(* First run program given in A072000 to define the SemiPrimePi function *) Table[SemiPrimePi[2^Prime[n]], {n, 10}](* Alonso del Arte, Dec 10 2010 *)

a(n)=my(N=2^prime(n),s,i); forprime(p=2, sqrtint(N), s+=primepi(N\p); i++); s - i * (i-1)/2 \\ Charles R Greathouse IV, Apr 25 2016

from math import isqrt
from sympy import prime, primepi
def A175613(n):
    m = 1<Chai Wah Wu, Jul 23 2024

a(n) = A072000(A034785(n)) = A125527(A000040(n)). - R. J. Mathar, Dec 10 2010

a(14) & a(15) from Robert G. Wilson v, Oct 19 2011.

A225533 Numbers expressible as p^2 + 2^q where p and q are primes.

Original entry on oeis.org

8, 12, 13, 17, 29, 33, 36, 41, 53, 57, 81, 125, 129, 132, 137, 153, 173, 177, 201, 249, 293, 297, 321, 365, 369, 393, 417, 489, 533, 537, 561, 657, 845, 849, 873, 965, 969, 993, 1089, 1373, 1377, 1401, 1497, 1685, 1689, 1713, 1809, 1853, 1857, 1881, 1977, 2052

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 09 2013

nonn

a(n) ~ n^2. For n > 468, the formula .358*n^2.085 provides an estimate of a(n) accurate to within 11%.

a(10) = 57 is the first term that meets the criterion in two ways (5^2 + 2^5 and 7^2 + 2^3). In the first 10000 terms, there are 30 terms expressible in two ways, but none expressible in three ways.

			29 = 5^2 + 2^2, and both 5 and 2 are prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Ulam Spiral of a(n) for n = 1..10000.
Christian N. K. Anderson, Table of n, a(n), and all values of (p,q) producing a(n) for n = 1..10000.

Cf. A000040, A001248, A034785.

nn = 15; ps = Prime[Range[nn]]; p2 = Prime[Range[PrimePi[2*Log[2, ps[[-1]]]]]]; t = Table[p^2 + 2^q, {p, ps}, {q, p2}]; Union[Select[Flatten[t], # < ps[[-1]]^2 &]] (* T. D. Noe, May 15 2013 *)

library(gmp); x=y=as.bigz(2); maxval=10000; sol=as.bigz(matrix(0,nc=3,nr=1000)); len=0
while(len<1000 & x^2+2^y

A243139 a(n) = 2^prime(n) + prime(n).

Original entry on oeis.org

6, 11, 37, 135, 2059, 8205, 131089, 524307, 8388631, 536870941, 2147483679, 137438953509, 2199023255593, 8796093022251, 140737488355375, 9007199254741045, 576460752303423547, 2305843009213694013, 147573952589676412995, 2361183241434822606919

Offset: 1

Vincenzo Librandi, Jun 03 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
J. Mestel, Archimedeans Problems Drive 1977, Eureka, 39 (1978), 38-40. (Annotated scanned copy)

Cf. A034785, A100105.

Magma
```
[2^p + p: p in PrimesUpTo(80)];
```

f[n_]:=(2^Prime[n] + Prime[n]); Array[f, 80, 1]

A280609 Odd prime powers with prime exponents.

Original entry on oeis.org

9, 25, 27, 49, 121, 125, 169, 243, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 16807, 17161, 18769, 19321, 22201, 22801, 24389, 24649, 26569, 27889, 29791, 29929

Offset: 1

Ilya Gutkovskiy, Jan 06 2017

Intersection of A053810 and A061345.

			9 is in the sequence because 9 = 3^2;
25 is in the sequence because 25 = 5^2;
27 is in the sequence because 27 = 3^3, etc.

Eric Weisstein's World of Mathematics, Prime Power.

Cf. A000961, A034785, A051006, A053810, A061345, A078422, A246547, A246551, A246655.

Select[Range[30000], PrimePowerQ[#1] && PrimeQ[PrimeOmega[#1]] && Mod[#1, 2] == 1 & ]

from sympy import primepi, integer_nthroot, primerange
def A280609(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0])-1 for p in primerange(x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

a(n) = p^q, where p, q are primes and p > 2.

Sum_{n>=1} 1/a(n) = Sum_{p prime} P(p) - A051006 = 0.25699271237062131298..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 13 2024

A319233 Numbers k such that k^2 + 1 divides 2^k + 4.

Original entry on oeis.org

0, 1, 8, 28, 32, 128, 2048, 8192, 23948, 131072, 524288, 8388608, 536870912, 2147483648, 137438953472

Offset: 1

Altug Alkan, Sep 14 2018

This sequence corresponds to numbers k such that k^2 + 1 divides 2^k + 2^m where m = 2 (A247220 (m = 0), A319216 (m = 1)).

a(16) > 10^12. - Hiroaki Yamanouchi, Sep 17 2018

			32 = 2^5 is a term since (2^(2^5) + 2^2)/((2^5)^2 + 1) = 2^22 - 2^12 + 2^2.

Cf. A034785, A247220, A319216.

PARI
```
isok(n)=Mod(2, n^2+1)^n==-4;
```

a(15) from Hiroaki Yamanouchi, Sep 17 2018

A333392 a(0) = 1; thereafter a(n) = 2^(prime(n)-1) + Sum_{k=1..n} 2^(prime(n)-prime(k)).

Original entry on oeis.org

1, 3, 7, 29, 117, 1873, 7493, 119889, 479557, 7672913, 491066433, 1964265733, 125713006913, 2011408110609, 8045632442437, 128730119078993, 8238727621055553, 527278567747555393, 2109114270990221573, 134983313343374180673, 2159733013493986890769, 8638932053975947563077

Offset: 0

Ilya Gutkovskiy, Mar 18 2020

nonn

			a(7) = 119889 (in base 10) = 11101010001010001 (in base 2).
                             ||| | |   | |   |
                             123 5 7  1113  17

Cf. A000040, A008578, A010051, A034785, A051006, A072762, A076793, A080339, A080355, A121240, A139104, A333393.

a[0] = 1; a[n_] := 2^(Prime[n] - 1) + Sum[2^(Prime[n] - Prime[k]), {k, 1, n}]; Table[a[n], {n, 0, 21}]

a(n) = if (n==0, 1, 2^(prime(n)-1) + sum(k=1, n, 2^(prime(n)-prime(k)))); \\ Michel Marcus, Mar 18 2020

a(n) = floor(c * 2^prime(n)) for n > 0, where c = 0.91468250985... = 1/2 + A051006.

A272260 Numbers that cause an infinite loop in Conway's PRIMEGAME.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 22, 25, 26, 29, 31, 33, 35, 37, 39, 41, 43, 44, 47, 49, 51, 52, 53, 55, 57, 59, 61

Offset: 1

Alonso del Arte, Apr 23 2016

The following values are certainly in the sequence: 65, 67, 71, 73, 77, 79, 83, 87, 88, 89, 91, 97, 99, 101. The following values are doubtful: 62, 74, 82, 86, 93, 94.
Conway's PRIMEGAME (also called "Conway's prime producing machine") is a fascinating (and very inefficient) method for obtaining the prime numbers.
The "machine" takes in a number, and tries multiplying it by each of fourteen fractions one by one to find the first one that produces an integer. Then that integer is multiplied by each of the fourteen fractions one by one to find the first one that produces another integer. The goal is to find powers of 2; these powers of 2 have a binary logarithm that is a prime number.
The fractions of Conway's PRIMEGAME are 17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/2, 1/7, 55.
The "machine" was designed to take 2 as its first input, which gives us the sequence A007542, and from that sequence we can pick out the sequence 2^prime(n) (A034785).
But there are other numbers that can be used as a first input. If the process is started with 3, the process eventually leads to 2 (see A185242). So starting with 3 just delays the process.
However, the numbers in this sequence taken as first inputs do much worse than delay the process, they get the program stuck in an endless loop.
A lot, but not all, of the numerators of the Conway fractions are in this sequence. Specifically, all except 78, 23, 95, 15. As for denominators, all of them except 85, 38, 23, 2 are in this sequence.
All prime numbers greater than 29 are in this sequence. Given a prime number p > 29, we see that multiplying by the first thirteen fractions results in a rational but non-integer value, so the process gives 55p for the first step. Then 55p * 13/11 = 65p and 65p * 11/13 = 55p, hence an infinite loop.
In fact, the only primes that can be used to start the process without leading to an infinite loop are 2, 3, 23.

			5 multiplied by 55 gives 275.
275 multiplied by 13/11 gives 325.
325 multiplied by 11/13 gives 275.
Since 275 has occurred before, this means that 5 leads the process to get stuck on bouncing between 275 and 325, and so 5 is in this sequence.

Cf. A203907.

A289898 a(n) = floor((2^prime(n+1))/Sum_{k=0|n,2^prime(k)}).

Original entry on oeis.org

2, 2, 2, 11, 3, 12, 3, 12, 59, 3, 51, 15, 3, 12, 59, 62, 3, 51, 15, 3, 50, 15, 60, 251, 15, 3, 12, 3, 12, 15179, 15, 60, 3, 816, 3, 51, 62, 15, 60, 62, 3, 816, 3, 12, 3, 3226, 4094, 15, 3, 12, 59, 3, 816, 63, 63, 63, 3, 51, 15, 3, 808, 16363, 15, 3, 12, 15183

Offset: 1

Joseph Wheat, Jul 14 2017

nonn

Cf. A034785, A076793.

Table[Floor[2^Prime[n + 1]/Sum[ 2^Prime[k], {k, n}]], {n, 66}] (* Michael De Vlieger, Jul 16 2017 *)

a(n) = 2^prime(n+1)\sum(k=1, n, 2^prime(k)); \\ Michel Marcus, Jul 16 2017

a(n) = floor(2^prime(n+1)/(Sum_{k=1..n} 2^prime(k))).

cp's OEIS Frontend

A175613 Number of semiprimes <= 2^prime(n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A225533 Numbers expressible as p^2 + 2^q where p and q are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

R

A243139 a(n) = 2^prime(n) + prime(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A280609 Odd prime powers with prime exponents.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A319233 Numbers k such that k^2 + 1 divides 2^k + 4.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A333392 a(0) = 1; thereafter a(n) = 2^(prime(n)-1) + Sum_{k=1..n} 2^(prime(n)-prime(k)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A272260 Numbers that cause an infinite loop in Conway's PRIMEGAME.

Views

Author

Keywords

Comments

Examples

Crossrefs

A289898 a(n) = floor((2^prime(n+1))/Sum_{k=0|n,2^prime(k)}).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI