A240233 a(n) is the smallest prime number such that both a(n) + 6n and a(n) + 12n are prime numbers.

5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 5, 7, 11, 5, 11, 5, 7, 23, 13, 11, 5, 5, 41, 5, 7, 37, 29, 11, 5, 13, 7, 5, 13, 23, 13, 7, 5, 5, 23, 11, 11, 5, 5, 13, 7, 5, 29, 23, 13, 7, 5, 19, 41, 13, 17, 11, 7, 5, 19, 7, 7, 7, 5, 5, 7, 5, 7, 11, 29, 13, 5, 17, 5, 19, 7, 7, 5

Offset

1,1

Comments

a(n), a(n) + 6n, and a(n) + 12n form an arithmetic progression with a common difference of 6n.

If the interval is not a multiple of six, such an arithmetic progression of primes cannot exist unless a(n)=3. For example, 3,5,7 has an interval of 2; 3,7,11 has an interval of 4; and 3,11,19 has an interval of 8, as in A115334 and A206037.

Conjecture: a(n) is defined for all n > 0.

Links

Lei Zhou, Table of n, a(n) for n = 1..10000

Example

n=1, 6n=6. 5,11,17 are all prime numbers with an interval of 6. So a(1)=5;
...
n=13, 6n=78. 5+78=83, 5+2*78=161=7*23(x); 7+78=85(x); 11+78=89, 11+78*2=167. 11,89,167 are all prime numbers with an interval of 78. So a(13)=11.

Mathematica

Table[diff = n*6; k = 1; While[k++; p = Prime[k]; cp1 = p + diff; cp2 = p + 2*diff; ! ((PrimeQ[cp1]) && (PrimeQ[cp2]))]; p, {n, 77}]

Crossrefs

Cf. A000040, A115334, A206037, A240087.

Sequence in context: A076568 A139259 A105444 * A033299 A293982 A071577

Adjacent sequences: A240230 A240231 A240232 * A240234 A240235 A240236

Keyword

nonn,easy

Author

Lei Zhou, Apr 02 2014

Status

approved