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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. 2
 4437, 67088885, 608096563245, 6008043480300405, 60017281285205688005, 600012360124320087600005, 6000055320121974202106400005, 60000010840001925680009488000005, 600000005880000160040000148000000005, 6000000035120000052560000001460000000005 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This is an interesting sequence because for most cases Δ<0. The cases where Δ>0 are sparse. 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. 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 The corresponding least palindromic primes are: 11, 929, 98689, 9989899, 999727999, 99999199999, 9999987899999, 999999787999999, ... 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 REFERENCES D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer. LINKS EXAMPLE 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. 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. The corresponding values of a,b,c are given in the table bellow. +----+---------------------+-----------------------+-----------------------+ |  n |          a          |           b           |            c          | +----+---------------------+-----------------------+-----------------------+ |  1 |                  11 |                   101 |                   131 | |  2 |                 929 |                 10301 |                 10501 | |  3 |               98689 |               1003001 |               1008001 | |  4 |             9989899 |             100030001 |             100050001 | |  5 |           999727999 |           10000500001 |           10000900001 | |  6 |         99999199999 |         1000008000001 |         1000017100001 | |  7 |       9999987899999 |       100000323000001 |       100000353000001 | |  8 |     999999787999999 |     10000000500000001 |     10000001910000001 | |  9 |   99999999299999999 |   1000000008000000001 |   1000000032300000001 | | 10 | 9999999992999999999 | 100000000212000000001 | 100000000252000000001 | +----+---------------------+-----------------------+-----------------------+ CROSSREFS Cf. A002385, A327447. See also A028990 and A028989. Sequence in context: A060363 A128194 A184091 * A206016 A091739 A166582 Adjacent sequences:  A309484 A309485 A309486 * A309488 A309489 A309490 KEYWORD nonn,base AUTHOR Philip Mizzi, Sep 06 2019 EXTENSIONS a(6)-a(8) from Daniel Suteu, Sep 08 2019 a(9) from Chai Wah Wu, Sep 09 2019 a(10) from Chai Wah Wu, Sep 12 2019 STATUS approved

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Last modified January 24 07:18 EST 2020. Contains 331189 sequences. (Running on oeis4.)