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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
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.
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
See also A028990 and A028989.
Sequence in context: A060363 A128194 A184091 * A206016 A091739 A166582
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