A238224 -id:A238224 - cp's OEIS frontend

A238278 a(n) = |{0 < k < n: the number of primes in the interval ((k-1)n, kn] and the number of primes in the interval (kn, (k+1)n] are both prime}|.

0, 0, 0, 1, 1, 3, 3, 2, 7, 6, 8, 4, 9, 4, 9, 8, 1, 1, 3, 5, 4, 6, 3, 4, 4, 6, 3, 11, 8, 8, 7, 7, 12, 9, 4, 8, 9, 12, 8, 12, 8, 7, 6, 7, 7, 9, 4, 8, 9, 11, 5, 6, 3, 11, 2, 5, 14, 8, 8, 11, 2, 1, 11, 4, 6, 4, 5, 4, 1, 9, 5, 2, 10, 5, 4, 9, 10, 11, 6, 7

Offset: 1

Zhi-Wei Sun, Feb 22 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) For any integer n > 3, there is a prime p < n such that the number of primes in the interval ((p-1)*n, p*n) is a prime.
We have verified part (i) for n up to 150000.
See also A238277 and A238281 for related conjectures.

			a(17) = 1 since the interval (9*17, 10*17] contains exactly 3 primes with 3 prime, and the interval (10*17, 11*17] contains exactly 3 primes with 3 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3500
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A237578, A237643, A237705, A238224, A238277, A238281.

d[k_,n_]:=PrimePi[k*n]-PrimePi[(k-1)n]
a[n_]:=Sum[If[PrimeQ[d[k,n]]&&PrimeQ[d[k+1,n]],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A238281 a(n) = |{0 < k < n: the two intervals (kn, (k+1)n) and ((k+1)n, (k+2)n) contain the same number of primes}|.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 3, 1, 5, 2, 4, 4, 8, 3, 7, 4, 4, 4, 2, 3, 7, 3, 10, 4, 12, 7, 7, 15, 7, 9, 8, 5, 8, 9, 11, 8, 8, 10, 8, 4, 10, 10, 10, 11, 7, 10, 8, 11, 8, 8, 9, 9, 8, 11, 7, 8, 13, 10, 8, 14, 13, 4, 14, 8, 11, 12, 14, 12, 8, 10, 16, 12, 16, 12, 14, 19, 11, 14, 8, 9

Offset: 1

Zhi-Wei Sun, Feb 22 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, if n > 1 is not equal to 8, then there is a positive integer k < n with 2*k + 1 prime such that the two intervals ((k-1)*n, k*n) and (k*n, (k+1)*n) contain the same number of primes.

(ii) For any integer n > 4, there is a positive integer k < prime(n) such that all the three intervals (k*n, (k+1)*n), ((k+1)*n, (k+2)*n), ((k+2)*n, (k+3)*n) contain the same number of primes, i.e., pi(k*n), pi((k+1)*n), pi((k+2)*n), pi((k+3)*n) form a 4-term arithmetic progression.

			a(8) = 1 since each of the two intervals (7*8, 8*8) and (8*8, 9*8) contains exactly two primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000720, A237578, A238224, A238277, A238278.

d[k_,n_]:=PrimePi[(k+1)*n]-PrimePi[k*n]
a[n_]:=Sum[If[d[k,n]==d[k+1,n],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A238165 Number of pairs {j, k} with 0 < j < k <= n such that pi(jn) divides pi(kn), where pi(.) is given by A000720.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 1, 5, 5, 5, 3, 5, 12, 5, 5, 7, 3, 2, 12, 7, 8, 9, 9, 6, 6, 11, 9, 12, 9, 15, 12, 12, 13, 7, 16, 12, 18, 15, 16, 11, 8, 8, 13, 15, 20, 13, 7, 15, 13, 7, 18, 7, 18, 15, 11, 15, 15, 12, 15, 17, 6, 18, 17, 16, 11, 15, 9, 18, 15, 13

Offset: 1

Zhi-Wei Sun, Feb 19 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) For any integer n > 4, the sequence pi(k*n)^(1/k) (k = 1, ..., n) is strictly decreasing.
See also A238224 for a refinement of part (i) of this conjecture.

			a(5) = 3 since pi(1*5) = 3 divides both pi(3*5) = 6 and pi(5*5) = 9, and pi(2*5) = 4 divides pi(4*5) = 8.
a(7) = 1 since pi(1*7) = 4 divides pi(3*7) = 8.

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

Cf. A000720, A237578, A237597, A237598, A237612, A237614, A237615, A238224.

m[k_,j_]:=Mod[PrimePi[k],PrimePi[j]]==0
a[n_]:=Sum[If[m[k*n,j*n],1,0],{k,2,n},{j,1,k-1}]
Do[Print[n," ",a[n]],{n,1,70}]

cp's OEIS Frontend

A238278 a(n) = |{0 < k < n: the number of primes in the interval ((k-1)n, kn] and the number of primes in the interval (kn, (k+1)n] are both prime}|.

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A238281 a(n) = |{0 < k < n: the two intervals (kn, (k+1)n) and ((k+1)n, (k+2)n) contain the same number of primes}|.

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A238165 Number of pairs {j, k} with 0 < j < k <= n such that pi(jn) divides pi(kn), where pi(.) is given by A000720.

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cp's OEIS Frontend

A238278 a(n) = |{0 < k < n: the number of primes in the interval ((k-1)*n, k*n] and the number of primes in the interval (k*n, (k+1)*n] are both prime}|.

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Programs

Mathematica

A238281 a(n) = |{0 < k < n: the two intervals (k*n, (k+1)*n) and ((k+1)*n, (k+2)*n) contain the same number of primes}|.

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Mathematica

A238165 Number of pairs {j, k} with 0 < j < k <= n such that pi(j*n) divides pi(k*n), where pi(.) is given by A000720.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A238278 a(n) = |{0 < k < n: the number of primes in the interval ((k-1)n, kn] and the number of primes in the interval (kn, (k+1)n] are both prime}|.

A238281 a(n) = |{0 < k < n: the two intervals (kn, (k+1)n) and ((k+1)n, (k+2)n) contain the same number of primes}|.

A238165 Number of pairs {j, k} with 0 < j < k <= n such that pi(jn) divides pi(kn), where pi(.) is given by A000720.