cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A238573 a(n) = |{0 < k <= n: prime(k*n) + 2 is prime}|.

Original entry on oeis.org

0, 1, 1, 0, 3, 0, 4, 1, 2, 3, 1, 3, 4, 4, 4, 4, 1, 3, 6, 5, 3, 3, 4, 6, 3, 8, 5, 6, 3, 4, 2, 10, 6, 5, 7, 8, 6, 8, 7, 5, 7, 5, 11, 7, 7, 8, 8, 11, 6, 5, 7, 11, 11, 7, 4, 9, 7, 3, 5, 7, 7, 11, 8, 13, 9, 8, 7, 7, 12, 10, 8, 11, 8, 15, 8, 9, 9, 15, 13, 4
Offset: 1

Views

Author

Zhi-Wei Sun, Mar 01 2014

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n > 6, and a(n) = 1 only for n = 2, 3, 8, 11, 17. Moreover, for any n > 0 there exists a positive integer k < 3*sqrt(n) + 6 such that prime(k*n) + 2 is prime.
(ii) For any integer n > 6, prime(k^2*n) + 2 is prime for some k = 1, ..., n.
(iii) If n > 5, then prime(k^2*(n-k)) + 2 is prime for some 0 < k < n.
Clearly, each of the three parts implies the twin prime conjecture.
We have verified part (i) of the conjecture for n up to 2*10^6.

Examples

			a(2) = 1 since prime(1*2) + 2 = 3 + 2 = 5 is prime.
a(3) = 1 since prime(1*3) + 2 = 5 + 2 = 7 is prime.
a(8) = 1 since prime(8*8) + 2 = 311 + 2 = 313 is prime.
a(11) = 1 since prime(3*11) + 2 = 137 + 2 = 139 is prime.
a(17) = 1 since prime(1*17) + 2 = 59 + 2 = 61 is prime.

Links

Crossrefs

Cf. A000040, A001359, A006512, A218829, A236531, A237578, A238576.

Programs

  • Mathematica
    p[k_,n_]:=PrimeQ[Prime[k*n]+2]
a[n_]:=Sum[If[p[k,n],1,0],{k,1,n}]
Table[a[n],{n,1,80}]

A238878 a(n) = |{0 < k <= n: prime(prime(k)) - prime(k) + 1 and prime(prime(k*n)) - prime(k*n) + 1 are both prime}|.

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Mar 06 2014

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 4, 5, 19, 77.
(ii) For any integer n > 0, there is a number k among 1, ..., n such that 2*k + 1 and prime(prime(k^2*n)) - prime(k^2*n) + 1 are both prime.

Examples

			a(5) = 1 since prime(prime(4)) - prime(4) + 1 = prime(7) - 7 + 1 = 17 - 6 = 11 and prime(prime(4*5)) - prime(4*5) + 1 = prime(71) - 71 + 1 = 353 - 70 = 283 are both prime.
a(77) = 1 since prime(prime(3)) - prime(3) + 1 = prime(5) - 5 + 1 = 11 - 4 = 7 and prime(prime(3*77)) - prime(3*77) + 1 = prime(1453) - 1453 + 1 = 12143 - 1452 = 10691 are both prime.

Links

Crossrefs

Cf. A000040, A234695, A237715, A238573, A238576, A238756, A238766, A238776, A238814, A238881.

Programs

  • Mathematica
    PQ[n_]:=PrimeQ[Prime[n]-n+1]
p[k_,n_]:=PQ[Prime[k]]&&PQ[Prime[k*n]]
a[n_]:=Sum[If[p[k,n],1,0],{k,1,n}]
Table[a[n],{n,1,80}]

A238881 Number of odd primes p < 2*n with prime(n*(p+1)/2) + n*(p+1)/2 prime.

Original entry on oeis.org

0, 1, 1, 0, 0, 2, 1, 3, 1, 2, 2, 5, 2, 3, 2, 4, 3, 5, 2, 3, 2, 4, 1, 8, 4, 4, 4, 3, 2, 5, 2, 4, 4, 4, 4, 8, 2, 4, 2, 5, 4, 6, 3, 2, 6, 6, 3, 11, 6, 10, 4, 8, 2, 11, 4, 7, 4, 7, 2, 12, 4, 6, 2, 6, 3, 8, 3, 5, 8, 12, 6, 12, 4, 15, 8, 11, 5, 12, 2, 11
Offset: 1

Views

Author

Zhi-Wei Sun, Mar 06 2014

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n > 5, and a(n) = 1 only for n = 2, 3, 7, 9, 23. Moreover, for any r = 1,-1 and n > 5*(2+r) there is a positive integer k < n such that 2*k+r and prime(k*n)+k*n are both prime.
(ii) If n > 1 is not equal to 13, then prime(k*n) - k*n is prime for some k = 1, ..., n.
This conjecture implies that there are infinitely many positive integers m with prime(m) + m (or prime(m) - m) prime.

Examples

			a(7) = 1 since 11 and prime(7*(11+1)/2) + 7*(11+1)/2 = prime(42) + 42 = 181 + 42 = 223 are both prime.
a(23) = 1 since 7 and prime(23*(7+1)/2) + 23*(7+1)/2 = prime(92) + 92 = 479 + 92 = 571 are both prime.

References

  • Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169--187. (See Conjecture 3.21(i) and note that the typo 2k+1 there should be 2k-1.)

Links

Crossrefs

Cf. A000040, A064269, A064402, A238573, A238576, A238878.

Programs

  • Mathematica
    PQ[n_]:=PrimeQ[Prime[n]+n]
p[k_,n_]:=PQ[(Prime[k]+1)/2*n]
a[n_]:=Sum[If[p[k,n],1,0],{k,2,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]
  • PARI
    a(n) = {my(nb = 0); forprime(p=3, 2*n, if (isprime(prime(n*(p+1)/2) + n*(p+1)/2), nb++);); nb;} \\ Michel Marcus, Sep 21 2015

A238890 a(n) = |{0 < k <= n: prime(k*n) - pi(k*n) is prime}|, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 4, 1, 2, 2, 3, 6, 1, 1, 4, 4, 1, 5, 3, 5, 5, 4, 5, 1, 2, 5, 7, 6, 5, 2, 2, 4, 4, 4, 10, 6, 5, 5, 4, 6, 8, 7, 5, 8, 5, 8, 5, 3, 5, 9, 6, 7, 2, 2, 4, 6, 7, 8, 11, 8, 8, 10, 6, 8, 10, 2, 5, 11, 7, 5, 10, 10, 8, 7, 9, 8
Offset: 1

Views

Author

Zhi-Wei Sun, Mar 06 2014

Keywords

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 28.
(ii) If n > 7 is not equal to 34, then prime(k*n) + pi(k*n) is prime for some k = 1, ..., n.
The conjecture implies that there are infinitely many primes p with p - pi(pi(p)) (or p + pi(pi(p))) prime.

Examples

			a(5) = 1 since prime(3*5) - pi(3*5) = 47 - 6 = 41 is prime.
a(28) = 1 since prime(18*28) - pi(18*28) = prime(504) - pi(504) = 3607 - 96 = 3511 is prime.

Links

Crossrefs

Cf. A000040, A000720, A237578, A237712, A238573, A238576, A238878, A238881.

Programs

  • Mathematica
    p[k_]:=PrimeQ[Prime[k]-PrimePi[k]]
a[n_]:=Sum[If[p[k*n],1,0],{k,1,n}]
Table[a[n],{n,1,80}]
Showing 1-4 of 4 results.

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