A056561 -id:A056561 - cp's OEIS frontend

A256302 Least prime p such that p+3k(k+1) is prime for all k=0,...,n.

2, 5, 5, 5, 11, 11, 11, 11, 11, 11, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23

Offset: 0

M. F. Hasler, Apr 05 2015

Inspired by the so-called prime-generating polynomials.

Since p | 3n(n+1) for n=p-1, one has a(n) > p(n). Otherwise stated, a(p-1) = p (as, e.g., for p=23) is optimal.

Cf. A181963, A181969, A181973, A210626, A211773, A212325, ...
Cf. A005846, A056561.
Cf. A256300, A256301.

a(n)=forprime(p=1,,for(i=1,n,ispseudoprime(p+3*(i+1)*i)||next(2));return(p))

A284043 Starts of a run of at least n consecutive numbers k for which k^2 - k + 41 is composite.

Original entry on oeis.org

41, 41, 122, 162, 299, 326, 326, 1064, 1064, 1064, 1064, 1064, 5664, 5664, 5664, 5664, 9265, 9265, 9265, 22818, 22818, 37784, 37784, 47494, 100202, 100202, 100202, 167628, 167628, 167628, 167628, 167628, 167628, 167628, 167628, 176956, 176956, 176956, 1081297

Offset: 1

Amiram Eldar, Jun 14 2017

nonn

This sequence is inspired by the problem proposed by Sidney Kravitz in 1963: "It is known that f(n)=n^2-n+41 yields prime numbers for n=1, 2, ..., 40. Find a sequence of 40 consecutive values of n for which f(n) is composite." Lawrence A. Ringenberg and others suggested the solution that starts at f(1)*f(2)*...*f(40)+1 (about 4.890... * 10^101). B. A. Hausmann suggested the smaller solution that starts at f(1)*f(2)*...*f(20)-19 (about 3.213... * 10^42). The smallest solution is a(40) = 1081297.

			The values of f(n)=n^2-n+41 at 122, 123 and 124 are: 14803 = 113*131, 15047 = 41*367 and 15293 = 41*373. This is the first case of 3 consecutive composite values, thus a(3) = 122.

Thomas Koshy, Elementary Number Theory with Applications, Academic Press, 2nd edition, 2007, Chapter 2, p. 147, exercise 50.

Amiram Eldar, Table of n, a(n) for n = 1..130 (terms below 10^10)
Sidney Kravitz, Problem 527, Mathematics Magazine, Vol. 36, No. 4 (1963), p. 264.
Lawrence A. Ringenberg et al., A Prime Generator, Solutions to Problem 527, Mathematics Magazine, Vol. 37, No. 2 (1964), pp. 122-123.

Cf. A002837, A005846, A007634, A056561, A145292, A202018.

f[n_] := n^2 - n + 41; a = PrimeQ[f[Range[1, 10^7]]]; b = Split[a]; c = Length /@ b; d = Accumulate[c]; nc = Length[c]; e = {}; For[len = 0, len < 100, len++; k = 2;  While[k <= nc && c[[k]] < len, k += 2]; If[k <= nc && c[[k]] >= len, ind = d[[k - 1]] + 1; e = AppendTo[e, ind]]]; e

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A256302 Least prime p such that p+3k(k+1) is prime for all k=0,...,n.

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A284043 Starts of a run of at least n consecutive numbers k for which k^2 - k + 41 is composite.

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cp's OEIS Frontend

A256302 Least prime p such that p+3*k*(k+1) is prime for all k=0,...,n.

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PARI

A284043 Starts of a run of at least n consecutive numbers k for which k^2 - k + 41 is composite.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A256302 Least prime p such that p+3k(k+1) is prime for all k=0,...,n.