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A108739 Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3.

%I A108739 #66 Feb 16 2025 08:32:58
%S A108739 3,30,6,80,12,450,894,3636,70756,97220,66768,300840,1623568,8436308
%N A108739 Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3.
%C A108739 This allows larger terms of A051254 (which triple in digits each entry) to be given. Like A051254, currently requires Riemann Hypothesis to show sequence continues.
%C A108739 Currently a(11)=66768 generates only a probable prime number. - _Arkadiusz Wesolowski_, May 28 2011
%C A108739 Likewise a(12) and a(13) generate only a probable prime numbers, as well as being conditional on a(11) and a(12) being proved primes. Minimality of a(12)-a(13) is exhaustively tested. - _Serge Batalov_, Aug 06 2013
%C A108739 a(14) = 8436308 is found by _Ryan Propper_ and _Serge Batalov_, Apr 29 2024, but a few remaining gaps below this value were being double-checked. The double-check is now complete (see GitHub link). - _Ryan Propper_ and _Serge Batalov_, May 24 2024.
%D A108739 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
%H A108739 Chris K. Caldwell, <a href="https://t5k.org/notes/proofs/A3n.html">Mills' Theorem - a generalization</a>.
%H A108739 Chris K. Caldwell, The List of Largest Known Primes, <a href="https://t5k.org/primes/page.php?id=77907">The 11th Mills' prime</a>
%H A108739 Chris K. Caldwell and Yuanyou Cheng, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html">Determining Mills' Constant and a Note on Honaker's Problem</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
%H A108739 Sergey Batalov, <a href="https://github.com/s-batalov/Mills_Primes">Mills' b(14) search report</a>
%H A108739 Henri & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=(((((((((%3F%2B450%29%5E3%2B%3F&amp;action=Search">PRP Records</a> and <a href="http://www.primenumbers.net/prptop/searchform.php?form=Mills%3F&amp;action=Search">PRP Records, search:Mills</a>
%H A108739 W. H. Mills, <a href="http://dx.doi.org/10.1090/S0002-9904-1947-08849-2">A prime-representing function</a>, Bull. Amer. Math. Soc., Vol. 53 (1947), p. 604.
%H A108739 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MillsConstant.html">Mills' Constant</a>
%H A108739 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MillsPrime.html">Mills' Prime</a>
%H A108739 E. M. Wright, <a href="https://doi.org/10.1112/jlms/s1-29.1.63">A class of representing functions</a>, J. London Math. Soc., Vol. 29 (1954) pp. 63-71.
%F A108739 b(1) = 2; b(n+1) = nextprime(b(n)^3); a(n) = b(n+1)-b(n)^3;
%e A108739 The Mills' primes (given in A051254) are 2, 2^3+3 = 11, (2^3+3)^3+30 = 11^3+30 = 1361, ((2^3+3)^3+30)^3+6 = 1361^3+6 = 2521008887, etc. The terms added at each step yield this sequence. They are the least positive integers which added to the cube of the preceding prime yield again a prime, cf. formula. - _M. F. Hasler_, Jul 22 2013
%t A108739 B[1] = 2; B[n_] := B[n] = NextPrime[B[n - 1]^3]; Table[B[n + 1] - B[n]^3, {n, 7}] (* _Robert Price_, Jun 09 2019 *)
%o A108739 (PARI) p=2; until(, np=nextprime(p^3); print1(np-p^3, ", "); p=np) \\ _Jeppe Stig Nielsen_, Apr 22 2020
%Y A108739 Cf. A051254, A051021.
%K A108739 more,nonn,hard
%O A108739 1,1
%A A108739 _Chris K. Caldwell_, Jun 22 2005
%E A108739 a(9)-a(11) from Caldwell and Cheng, Aug 29 2005
%E A108739 Corrected by _T. D. Noe_, Sep 24 2008
%E A108739 a(12) (which generates a PRP) from _Serge Batalov_, Jul 19 2013
%E A108739 a(13) (which generates a PRP) from _Serge Batalov_, Aug 06 2013
%E A108739 a(14) (which generates a PRP) from _Ryan Propper_ and _Serge Batalov_, May 24 2024