A035250 -id:A035250 - cp's OEIS frontend

A298071 Number of primes between floor(3n/2) and 2n (inclusive).

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 4, 5, 6, 6, 6, 7, 6, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 7, 6, 7, 7, 7, 6, 6, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 8, 9, 8, 8, 9, 10, 9

Offset: 1

Bruno Berselli, Jan 11 2018

nonn

Cf. A000040, A035250, A056171.

a[n_] := PrimePi[2*n] - PrimePi[Floor[3*n/2]] + If[PrimeQ[Floor[ 3*n/2]], 1, 0]; Array[a, 100] (* Jean-François Alcover, Jan 11 2018 *)

A298071 = lambda n: len([p for p in (3*n//2..2*n) if is_prime(p)])
print([A298071(n) for n in (1..97)]) # Peter Luschny, Jan 11 2018

A066665 a(n) = #{(x,y) | 0<=y<=x<=n and x+y is prime}.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 14, 16, 19, 23, 27, 31, 35, 38, 42, 47, 52, 56, 61, 65, 70, 76, 82, 88, 94, 100, 107, 114, 121, 128, 136, 143, 150, 158, 166, 175, 185, 194, 203, 213, 223, 233, 243, 252, 262, 272, 282, 291, 301, 311, 322, 334, 346

Offset: 0

Reinhard Zumkeller, Jan 07 2002

nonn

			a(3)=5, as sums of (1,1), (2,0), (2,1), (3,0) and (3,2) give 5 primes.

Cf. A000720, A035250.

a(0) = 0, for n>0: a(n) = a(n-1) + A035250(n).

A074810 Numbers k such that the number of primes between k and 2k (inclusive) = largest prime factor of k.

Original entry on oeis.org

1, 2, 4, 8, 9, 28, 65, 114, 174, 186, 246, 623, 1784, 1832, 1912, 5121, 13810, 14090, 39413, 40403, 808822, 809858, 810026, 2201505, 2202735, 6047408, 6048656, 16463939, 16467271, 16472371, 121482371, 121495747, 330358060, 898100679

Offset: 1

Jason Earls, Sep 08 2002

nonn

			28 is a term because there are 7 primes between n = 28 and 2n = 56: 29, 31, 37, 41, 43, 47, 53; and the largest prime dividing 28 is 7.

Cf. A006530, A035250.

with(numtheory): a:=proc(n) if pi(2*n)-pi(n-1)=max(1, factorset(n)) then n fi end: seq(a(n), n=1..1000); # Emeric Deutsch, Feb 05 2006

More terms from Emeric Deutsch, Feb 05 2006

a(24)-a(34) from Donovan Johnson, Apr 23 2010

A078754 Numbers k such that for all m>k there are more than n primes between m and 2m (inclusive).

Original entry on oeis.org

1, 8, 14, 20, 21, 26, 33, 35, 48, 50, 51, 63, 74, 75, 81, 86, 90, 111, 114, 116, 119, 120, 128, 134, 140, 153, 155, 168, 174, 183, 186, 200, 204, 209, 215, 216, 219, 230, 243, 245, 249, 284, 285, 288, 296, 299, 300, 303, 320, 321, 323, 326, 329, 338, 354, 363

Offset: 1

Jason Earls, Jan 08 2003

nonn

The terms shown are only conjectured values.

			a(3)=14 because for m>14, more than 3 primes always exist between m and 2m (inclusive).

Cf. A035250.

pn2n(n) = sum(k=n,2*n,if(isprime(k),1,0));
mpn2n(a,m, w)=local(k,M); for(n=a,m, M=0; k=0; while(k

A220095 n such that there are no primes between n - sqrt(n) and n.

Original entry on oeis.org

1, 2, 11, 29, 125, 126, 127

Offset: 1

Jon Perry, Dec 04 2012

Conjecture: This sequence is complete.

Cf. A035250.

function isprime(i) {
var i, j;
if (i == 1) return false;
if (i == 2) return true;
if (i % 2 == 0) return false;
for (j = 3; j <= Math.floor(Math.sqrt(i)); j += 2)
if (i % j == 0) return false;
return true;
}
for (n = 1; n < 100000; n++) {
for (k = Math.ceil(n - Math.sqrt(n)); k < n; k++) {
ip = false;
if (isprime(k)) {ip = true; break;}
}
if (!ip) document.write(n + ", ");
}

Select[Range[1000], PrimePi[# - 1] == PrimePi[# - Sqrt[#]] &] (* Alonso del Arte, Dec 04 2012 *)

A353946 a(n) = (pi(2n-1) - pi(n-1))^pi(n) for n > 1, a(1) = 0.

Original entry on oeis.org

0, 2, 4, 4, 8, 8, 81, 16, 81, 256, 1024, 1024, 4096, 729, 4096, 15625, 78125, 16384, 390625, 65536, 390625, 1679616, 10077696, 10077696, 10077696, 10077696, 40353607, 40353607, 282475249, 282475249, 8589934592, 1977326743, 1977326743, 8589934592, 8589934592, 31381059609, 1000000000000

Offset: 1

Wesley Ivan Hurt, May 12 2022

nonn

Number of functions from P to Q, where P is the set of primes <= n and Q is the set of primes q such that n <= q <= 2n-1.

Cf. A000720 (pi), A035250, A352749.

Join[{0}, Table[(PrimePi[2 n - 1] - PrimePi[n - 1])^PrimePi[n], {n, 2, 30}]]

a(n) = A035250(n)^A000720(n) for n > 1, a(1) = 0.

cp's OEIS Frontend

A298071 Number of primes between floor(3*n/2) and 2*n (inclusive).

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Sage

A066665 a(n) = #{(x,y) | 0<=y<=x<=n and x+y is prime}.

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A074810 Numbers k such that the number of primes between k and 2k (inclusive) = largest prime factor of k.

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A078754 Numbers k such that for all m>k there are more than n primes between m and 2m (inclusive).

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PARI

A220095 n such that there are no primes between n - sqrt(n) and n.

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Mathematica

A353946 a(n) = (pi(2n-1) - pi(n-1))^pi(n) for n > 1, a(1) = 0.

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A298071 Number of primes between floor(3n/2) and 2n (inclusive).