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A309487 Positive integers represented by the quadratic form (the discriminant form) Δ = b^2 - 4ac, where a,b,c are consecutive palindromic primes.

%I A309487 #40 Sep 14 2019 20:29:53
%S A309487 4437,67088885,608096563245,6008043480300405,60017281285205688005,
%T A309487 600012360124320087600005,6000055320121974202106400005,
%U A309487 60000010840001925680009488000005,600000005880000160040000148000000005,6000000035120000052560000001460000000005
%N A309487 Positive integers represented by the quadratic form (the discriminant form) Δ = b^2 - 4ac, where a,b,c are consecutive palindromic primes.
%C A309487 This is an interesting sequence because for most cases Δ<0. The cases where Δ>0 are sparse.
%C A309487 Based on a study of Δ for the case when a,b,c are consecutive primes I conjecture (but have no proof) that now Δ is always negative.
%C A309487 The conjecture in the previous comment is true. It says p(n)^2 <= 4*p(n-1)*p(n+1), and this follows from p(n)^2 <= 4*p(n-1)*p(n), i.e. p(n) <= 4*p(n-1), which is true (see A327447, also Mitrinovic, Sect. VII.18 (b)). - _N. J. A. Sloane_, Sep 10 2019
%C A309487 The corresponding least palindromic primes are: 11, 929, 98689, 9989899, 999727999, 99999199999, 9999987899999, 999999787999999, ...
%C A309487 Apart from the first term, it appears that the values of "a" and "b" are given by A028990 and A028989, respectively. - _Daniel Suteu_, Sep 08 2019
%D A309487 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer.
%e A309487 Consecutive palindromic primes begin with 2,3,5. For a=2, b=3, c=5, Δ=b^2-4ac=-31. Since Δ<0 this is not a member of the sequence.
%e A309487 With consecutive palindromic primes 11,101,131 and a=11, b=101, c=131, Δ=b^2-4ac=4437, the first member of the sequence.
%e A309487 The corresponding values of a,b,c are given in the table bellow.
%e A309487 +----+---------------------+-----------------------+-----------------------+
%e A309487 |  n |          a          |           b           |            c          |
%e A309487 +----+---------------------+-----------------------+-----------------------+
%e A309487 |  1 |                  11 |                   101 |                   131 |
%e A309487 |  2 |                 929 |                 10301 |                 10501 |
%e A309487 |  3 |               98689 |               1003001 |               1008001 |
%e A309487 |  4 |             9989899 |             100030001 |             100050001 |
%e A309487 |  5 |           999727999 |           10000500001 |           10000900001 |
%e A309487 |  6 |         99999199999 |         1000008000001 |         1000017100001 |
%e A309487 |  7 |       9999987899999 |       100000323000001 |       100000353000001 |
%e A309487 |  8 |     999999787999999 |     10000000500000001 |     10000001910000001 |
%e A309487 |  9 |   99999999299999999 |   1000000008000000001 |   1000000032300000001 |
%e A309487 | 10 | 9999999992999999999 | 100000000212000000001 | 100000000252000000001 |
%e A309487 +----+---------------------+-----------------------+-----------------------+
%Y A309487 Cf. A002385, A327447.
%Y A309487 See also A028990 and A028989.
%K A309487 nonn,base
%O A309487 1,1
%A A309487 _Philip Mizzi_, Sep 06 2019
%E A309487 a(6)-a(8) from _Daniel Suteu_, Sep 08 2019
%E A309487 a(9) from _Chai Wah Wu_, Sep 09 2019
%E A309487 a(10) from _Chai Wah Wu_, Sep 12 2019