

A240233


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


1



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
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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



