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A134996
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Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes.
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5
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2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081, 188011, 188801, 1008001, 1022201, 1028011, 1055501, 1058011, 1082801, 1085801, 1088081, 1108201, 1108501, 1110881, 1120121, 1120211
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refs;
listen;
history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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The digits of a(n) are restricted to 0, 1, 2, 5, 8. - Ivan N. Ianakiev, Oct 08 2015
The first term containing all the possible digits is 108225151. There are 2958 such terms up to 10^12, the last one in this range being 188885250551. - Giovanni Resta, Oct 08 2015
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LINKS
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EXAMPLE
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120121 is such a number because 120121, 121021 (upside down), 151051 (mirror) and 150151 are all prime. (This is the smallest one in which all four numbers are distinct.)
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MATHEMATICA
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lst1={2, 5};
startQ[n_]:=First[IntegerDigits[n]]==1;
subQ[n_]:=Module[{lst={0, 1, 2, 5, 8}}, SubsetQ[lst, Union[IntegerDigits[n]]]];
rev[n_]:=Reverse[IntegerDigits[n]];
updown[n_]:=FromDigits[rev[n]];
mirror[n_]:=FromDigits[rev[n]/.{2-> 5, 5-> 2}];
updownmirror[n_]:=FromDigits[rev[mirror[n]]];
lst2=Select[Range@188801, And[startQ[#], subQ[#], PrimeQ[#], PrimeQ[updown[#]], PrimeQ[mirror[#]], PrimeQ[updownmirror[#]]]&];
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PROG
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(Python)
from sympy import isprime
from itertools import count, islice, product
def t(s): return s.translate({ord("2"):ord("5"), ord("5"):ord("2")})
def ok(s): # s is a string of digits
return all(isprime(int(w)) for w in [s, s[::-1], t(s), t(s[::-1])])
def agen(): # generator of terms
yield from (2, 5)
for d in count(2):
for mid in product("01258", repeat=d-2):
s = "1" + "".join(mid) + "1"
if ok(s): yield int(s)
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CROSSREFS
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KEYWORD
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nonn,base,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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