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.

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A109928 Least k such that there are n primes of the form kr+1 with 1<=r<=k. Index of the first occurrence of n in A085053.

Original entry on oeis.org

1, 2, 4, 13, 6, 10, 20, 47, 22, 18, 28, 34, 51, 40, 62, 36, 30, 50, 48, 74, 42, 82, 80, 54, 105, 72, 60, 147, 78, 66, 116, 134, 84, 96, 142, 146, 108, 90, 114, 102, 172, 206, 130, 160, 226, 120, 144, 138, 126, 196, 248, 262, 162, 170, 302, 156, 186, 274, 356, 174
Offset: 1

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Author

Amarnath Murthy, Jul 17 2005

Keywords

Examples

			a(6) = 10 as 10 is the least number that gives 6 primes of the form 10k +1, k<=10. (11,31,41,61,71,101).

Crossrefs

Cf. A085053 (number of primes of the form kn+1, with k<=n).

Programs

  • Mathematica
    t=Table[Length[Select[Range[n], PrimeQ[n#+1]&]], {n, 1000}]; Table[First[Flatten[Position[t, n]]], {n, 100}] (Noe)

Extensions

More terms from T. D. Noe, Jul 17 2005

A230243 Number of primes p < n with 3*p + 8 and (p-1)*n + 1 both prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 1, 4, 2, 2, 4, 2, 3, 2, 4, 3, 4, 4, 2, 2, 2, 1, 5, 3, 4, 3, 3, 2, 3, 4, 2, 2, 4, 2, 4, 4, 1, 5, 3, 2, 6, 4, 1, 5, 6, 3, 3, 5, 1, 5, 5, 2, 7, 5, 3, 4, 4, 3, 4, 6, 3, 4, 6, 4, 5, 6, 3, 7, 4, 2, 6, 1, 3, 5, 9, 3, 3, 7, 4, 3, 7, 1, 6, 5, 5, 5, 6, 3, 6, 7
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 13 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 4.
This implies A. Murthy's conjecture (cf. A034693) that for any integer n > 1, there is a positive integer k < n such that k*n + 1 is prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 21 2023

Examples

			a(8) = 1 since 8 = 3 + 5 with 3, 3*3+8 = 17, (3-1)*8+1 = 17 all prime.
a(17) = 1 since 17 = 7 + 10, and 7, 3*7+8 = 29, (7-1)*17+1 = 103 are all prime.

Links

Crossrefs

Cf. A034693, A085053, A086685, A023210, A219864, A230217, A230219, A230241.

Programs

  • Mathematica
    a[n_]:=Sum[If[PrimeQ[3Prime[i]+8]&&PrimeQ[(Prime[i]-1)n+1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]
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