A247834 -id:A247834 - cp's OEIS frontend

A247835 Indices of prime(n)^3 in A247834, or a(n)=0 if prime(n)^3 is not in A247834.

1, 0, 3, 4, 7, 9, 13, 15, 21, 27, 29, 37, 43, 47, 52, 61, 71, 74, 83, 89, 94, 105, 111, 123, 138, 145, 149, 158, 161, 168, 196, 208, 220, 226, 246, 248, 261, 276, 287, 299, 316, 319, 340, 345, 358, 364, 392, 422, 432, 436, 447, 464, 470, 496, 512, 530, 544, 549

Offset: 1

Vladimir Shevelev, Sep 24 2014

nonn

Conjecture: all a(n)>0, except for n=2.

			Using the formula, let us find the position in A247834, in which should be 17^3, if 17^3 belongs to A247834. Since 17 = prime(7), then we have a(7) = pi(17^(3/2)) - 6 = pi(70) - 6 = 13. Indeed, A247834(13) = 4913 = 17^3.

Cf. A030078, A156759, A247393, A247394, A247395, A247396, A247509, A247510, A247511, A247606, A247834, A247867.

a(n) = my(p=prime(n)); if(nextprime(ceil(p*sqrt(p))) > p*sqrt(prime(n+1)), 0, primepi(prime(n)^(3/2)) - n + 1); \\ Jinyuan Wang, Feb 17 2021

If prime(n)^3 is in A247834, then a(n) = pi(prime(n)^(3/2)) - n + 1, where pi(x) is the prime counting function (A000720).

More terms from Jinyuan Wang, Feb 17 2021

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.

Original entry on oeis.org

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

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

A247835 Indices of prime(n)^3 in A247834, or a(n)=0 if prime(n)^3 is not in A247834.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

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

cp's OEIS Frontend

A247835 Indices of prime(n)^3 in A247834, or a(n)=0 if prime(n)^3 is not in A247834.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

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.