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Indices k such that A139250(k) = A000979(n).
3

%I #67 Apr 03 2023 10:36:12

%S 2,4,8,32,64,256,512,2048,32768,2097152,1073741824,549755813888,

%T 1125899906842624,9223372036854775808,9671406556917033397649408,

%U 39614081257132168796771975168,633825300114114700748351602688

%N Indices k such that A139250(k) = A000979(n).

%C Indices k such that the number of toothpicks in the toothpick structure of A139250 after k-th stage equals the n-th Wagstaff prime A000979. All terms of this sequence are powers of 2 (see formulas).

%C For a picture of the n-th Wagstaff prime as a toothpick structure see the Applegate link "A139250: the movie version", then enter N = a(n) and click "Update", for N = a(n) <= 32768 (due to the resolution of the movie).

%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.]

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="http://arxiv.org/abs/1004.3036">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, arXiv:1004.3036 [math.CO], 2010.

%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>

%H C. Caldwell's The Top Twenty, <a href="https://t5k.org/top20/page.php?id=67">Wagstaff</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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wagstaff_prime">Wagstaff prime</a>

%F a(n) = 2^A127936(n) = 2^(floor(A000978(n)/2)) = 2^(ceiling(log_4(A000979(n)))).

%F A139250(2^n) = A007583(n), n >= 0.

%F A139250(a(n)) = A000979(n).

%e For n = 5 we have that a(5) = 64, then we can see that the number of toothpicks in the toothpick structure of A139250 after 64 stages is 2731 which coincides with the fifth Wagstaff prime, so we can write A139250(64) = A000979(5) = 2731. See the illustration in the Applegate-Pol-Sloane paper, figure 3: T(64) = 2731 toothpicks.

%t 2^Reap[Do[If[PrimeQ[1+Sum[2^(2n-1), {n, m}]], Sow[m]], {m, 100}]][[2, 1]] (* _Jean-François Alcover_, Oct 06 2018 *)

%Y Cf. A000978, A000979, A007583, A001045, A127936, A127958, A127962, A127965, A139250, A139253.

%K nonn

%O 1,1

%A _Omar E. Pol_, Oct 23 2011

%E More terms from _Omar E. Pol_, Mar 14 2012