[seqfan] Strobogrammatic primes in base 1<k<11
Jonathan Post
jvospost3 at gmail.com
Mon Jan 26 00:54:03 CET 2009
I'm wondering how best to present an array of integers which are, in
one sense, merely "primes in A155584", and in another sense,
nontrivial "Strobogrammatic" primes, i.e. which are the same
"upside-down" in bases more than one and no more than 10.
I have to say "nontrivial" because every nonnegative integer is
strobogrammatic base 1.
Strobogrammatic binary primes == primes in A006995 == A016041.
Strobogrammatic primes base 3 = 13, 757, 1093, 9103, ... == primes
strobogrammatic in bases 2 and 3.
For bases 2 < k < 8 we have that every strobogrammatic prime base k
must also be strobogrammatic base 2 and hence palindromatic base 2
(obvious proof left to readers).
Hence we have, for example, strobogrammatic base 4 primes = A056130 =
"Palindromic primes in bases 2 and 4." This has a nice Mathematica
code by
Robert G. Wilson v, which anyone with Mathematica (I don't yet) can
tweak to check and extend the results that I show in this email.
Strobogrammatic primes base 5 = 31, 19531, 394501, 472631, ... ==
primes strobogrammatic in base 2 and base 5.
Strobogrammatic primes base 6 = 7, 37, 43, 1297, 55987, ... == primes
strobogrammatic in base 2 and base 6.
Note that 1101011 (base 6) = 18881 (base 10) which is strobogrammatic
base 10 but not prime base 6 nor 10 (though prime base 2).
Strobogrammatic primes base 7 = 2801, 134807, this last being
strobogrammatic prime in bases 2, 4, and 7.
Strobogrammatic primes base 8 = 73, 262657, 295433, ...
Strobogrammatic primes base 9 break the above pattern, as the can have
the digit 8, and are A068188 (tetradic primes).
Strobogrammatic primes base 10 == A007597.
We have the usual result that, except sometimes for the first element,
these (for the same range of k) must all have an odd number of digits.
Again, how best to present this in OEIS? Merely listing those numbers
which are strobogrammatic in some base 1 < k < 11 loses so much
structure.
3, 5, 7, 11, 13, 17, 31, 37, 43, 73, 101, 107, 127, ...
Showing the array by antidiagonals seems forced to me.
There is also A133207 Strobogrammatic non-palindromic primes.
Thanks for your help.
-- Jonathan Vos Post
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