%I #39 Apr 03 2023 10:36:11

%S 3,7,11,23,43,47,67,79,223,251,283,347,383,571,683,719,859,1163,1279,

%T 1319,1607,1667,1759,1871,2011,2731,2767,2843,2879,3919,4523,4783,

%U 4799,4903,5051,6343,6607,7823,8971,9643,11003,11071,11483,11519,12251,12907

%N Toothpick primes: primes in the toothpick sequence A139250.

%C All Wagstaff primes A000979 are members of this sequence. - _Omar E. Pol_, Mar 12 2012

%H R. J. Mathar, <a href="/A139253/b139253.txt">Table of n, a(n) for n = 1..181</a>

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], <a href="http://arxiv.org/abs/1004.3036">arXiv:1004.3036</a>

%H Chris K. Caldwell and G. L. Honaker, Jr., <a href="https://t5k.org/curios/cpage/23065.html">Prime Curios! 2011</a>

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>

%F A000040 INTERSECT A139250.

%e a(25) = 2011 because 2011 is the number of toothpicks after 60 stages in the toothpick structure of A139250 and it is also a prime number A000040.

%t toothpicks = With[{terms = 200}, CoefficientList[(x/((1-x)(1+2x)))(1 + 2x Product[1 + x^(2^k-1) + 2 x^(2^k), {k, 0, Log[2, terms] // Ceiling}]) + O[x]^terms, x]];

%t Select[toothpicks, PrimeQ] (* _Jean-François Alcover_, Aug 05 2018 *)

%Y Intersection of A000040 and A139250.

%Y Cf. A000979, A139251, A139252, A194810.

%K nonn

%O 1,1

%A _Omar E. Pol_, Apr 24 2008

%E More terms from _Omar E. Pol_, Dec 20 2008

%E Terms beyond 2731 from _R. J. Mathar_, Jan 21 2010