A247835 -id:A247835 - cp's OEIS frontend

A247867 a(n) is the smallest prime in the interval [ksqrt(k), ksqrt(k+2)], where k = A001359(n), or a(n)=0 if there is no prime in this interval.

0, 13, 37, 71, 157, 263, 457, 599, 1019, 1109, 1607, 1823, 2399, 2647, 2767, 3433, 3697, 4421, 4721, 5501, 6469, 8581, 8951, 9901, 11897, 13577, 14669, 15329, 16229, 16921, 23011, 23531, 23789, 25097, 26153, 32531, 33107, 33997, 34583, 36037, 39079, 43093

Offset: 1

Vladimir Shevelev, Sep 25 2014

nonn

The sequence is partly connected with conjecture in A247834. In turn, we conjecture that all terms a(n)>0 for n>1.

			For n=1, k=A001359(1)=3, we have the interval [3*sqrt(3), 3*sqrt(5)] = [5.1...,6.7...] which does not contain a prime. So, a(1)=0.
For n=2, k=5, we have the interval [5*sqrt(5), 5*sqrt(7)] = [11.1..., 13.2...] which contains only one prime: 13. So, a(2)=13.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001359, A247834, A247835.

p:= 1: q:= 2: count:= 0:
while count < 100 do
  if q = p+2 then
    count:= count+1;
    r:= nextprime(floor(p*sqrt(p)));
    if r^2 < p^2*q then A[count]:= r
    else A[count]:= 0 fi;
  fi;
  p:= q; q:= nextprime(p);
od:
seq(A[i],i=1..100); # Robert Israel, Apr 08 2018

lista(nn) = {forprime(p=2, nn, if (isprime(q=p+2), pmin = nextprime(ceil(p*sqrt(p))); if (pmin <= floor(p*sqrt(q)), val = pmin, val = 0); print1(val, ", ");););} \\ Michel Marcus, Sep 25 2014

More terms from Michel Marcus, Sep 25 2014

A247834 Maximal non-semiprime number which is a "preprime" of the n-th kind (defined in comment in A247395).

Original entry on oeis.org

8, 45, 125, 343, 325, 833, 1331, 1573, 2197, 2057, 3211, 3289, 4913, 4901, 6859, 6647, 8303, 10051, 10469, 11191, 12167, 15341, 16399, 17081, 18259, 22103, 24389, 26071, 29791, 27347, 31117, 35557, 36163, 36859, 39401, 42439, 50653, 50933, 52111, 56129, 56699

Offset: 1

Vladimir Shevelev, Sep 24 2014

nonn

Conjecture: the sequence contains all cubes of primes, except for 3^3 (cf. A030078).
Prime(n)^3 is in the sequence iff the interval [prime(n)^(3/2), prime(n)*sqrt(prime(n+1))] contains a prime.
A simple algorithm for finding the position k=k(n) for which a(k) = prime(n)^3 is given in A247835 (see formula and example there).
Conjecture: every term has the form a(n)= p*q*r, where p<=q<=r are primes.

Vladimir Shevelev , A classification of the positive integers over primes

Cf. A030078, A156759, A247393, A247394, A247395, A247396, A247509, A247510, A247511, A247606, A247835, A247867.

More terms from Peter J. C. Moses, Sep 24 2014

A247977 If n = 1 or prime, then a(n) = 0; otherwise, if n is a preprime of k-th kind, then a(n) = k.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 0, 2, 1, 2, 0, 2, 0, 2, 1, 2, 0, 2, 1, 3, 2, 3, 0, 3, 0, 3, 2, 3, 1, 3, 0, 3, 2, 3, 0, 3, 0, 3, 2, 3, 0, 3, 1, 4, 3, 4, 0, 4, 2, 4, 3, 4, 0, 4, 0, 4, 3, 4, 2, 4, 0, 4, 3, 4, 0, 4, 0, 4, 3, 4, 1, 4, 0, 4, 3, 4, 0, 4, 2, 4, 3, 4, 0, 4, 1, 4, 3, 4, 2, 4, 0, 4, 3, 4, 0, 4, 0, 4, 3, 4, 0, 4, 0, 4, 3, 4, 0, 4, 2, 4, 3, 4, 1, 4, 1, 5, 4, 5, 3

Offset: 1

Vladimir Shevelev, Sep 28 2014

Preprimes of k-th kind are defined in comment in A247395.

			If n = 15, then, by the formula, we have a(15) = 2 - 2 + 1 = 1.

Indranil Ghosh, Table of n, a(n) for n = 1..10000
Vladimir Shevelev , A classification of the positive integers over primes

Cf. A000040, A156759, A247393, A247394, A247395, A247396, A247509, A247510, A247511, A247606, A247834, A247835.

Table[If[n==1 || PrimeQ[n], 0, PrimePi[Sqrt[n]] - PrimePi[FactorInteger[n][[1, 1]]] + 1], {n, 1, 125}] (* Indranil Ghosh, Mar 08 2017 *)

for(n=1, 125, print1(if(n==1 || isprime(n), 0, primepi(sqrt(n)) - primepi(vecmin(factor(n)[, 1])) + 1),", ")) \\ Indranil Ghosh, Mar 08 2017

If n is a composite number, then a(n) = pi(sqrt(n)) - pi(lpf(n)) + 1, where pi(x) is prime counting function (cf. A000720), lpf = least prime factor (A020639).

cp's OEIS Frontend

A247867 a(n) is the smallest prime in the interval [ksqrt(k), ksqrt(k+2)], where k = A001359(n), or a(n)=0 if there is no prime in this interval.

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A247834 Maximal non-semiprime number which is a "preprime" of the n-th kind (defined in comment in A247395).

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Author

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Extensions

A247977 If n = 1 or prime, then a(n) = 0; otherwise, if n is a preprime of k-th kind, then a(n) = k.

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Mathematica

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cp's OEIS Frontend

A247867 a(n) is the smallest prime in the interval [k*sqrt(k), k*sqrt(k+2)], where k = A001359(n), or a(n)=0 if there is no prime in this interval.

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Maple

PARI

Extensions

A247834 Maximal non-semiprime number which is a "preprime" of the n-th kind (defined in comment in A247395).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A247977 If n = 1 or prime, then a(n) = 0; otherwise, if n is a preprime of k-th kind, then a(n) = k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A247867 a(n) is the smallest prime in the interval [ksqrt(k), ksqrt(k+2)], where k = A001359(n), or a(n)=0 if there is no prime in this interval.