This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
import Data.List (unfoldr)
a048890 n = a048890_list !! (n-1)
a048890_list = filter f a000040_list where
f x = all (`elem` [0,1,6,8,9]) ds && x' /= x && a010051 x' == 1
where x' = foldl c 0 ds
c v 6 = 10*v + 9; c v 9 = 10*v + 6; c v d = 10*v + d
ds = unfoldr d x
d z = if z == 0 then Nothing else Just $ swap $ divMod z 10
-- Reinhard Zumkeller, Nov 18 2011
from sympy import isprime
from itertools import product
def ud(s):
return s[::-1].translate({ord('6'):ord('9'), ord('9'):ord('6')})
def auptod(maxdigits):
alst = []
for d in range(1, maxdigits+1):
for p in product("01689", repeat=d-1):
if d > 1 and p[0] == "0": continue
for end in "19":
s = "".join(p) + end
t, udt = int(s), int(ud(s))
if isprime(t) and isprime(udt): alst.append(t)
return alst
print(auptod(5)) # Michael S. Branicky, Nov 19 2021
16661 is a term because it is a prime and a palindrome as well; when rotated by 180 degrees it becomes 19991 that is also a prime.
Select[ lst = {}; fQ[n_] := Block[{allset = {0, 1, 6, 8, 9}, id = IntegerDigits@n}, rid = Reverse[id /. {6 -> 9, 9 -> 6}];Union@Join[id, allset] == allset && PrimeQ@FromDigits@rid && rid != id]; Do[If[PrimeQ@n && fQ@n, AppendTo[lst, n]], {n, 1090000000}]; lst, # ==FromDigits[Reverse[IntegerDigits[#]]] &]
is_palandinv(n) = my(d=digits(n), ineligible_d=[2, 3, 4, 5, 7]); d==Vecrev(d) && #setintersect(vecsort(d), ineligible_d)==0 invert(n) = my(d=digits(n), e=[]); for(k=1, #d, if(d[k]==0, e=concat(e, [0])); if(d[k]==1, e=concat(e, [1])); if(d[k]==6, e=concat(e, [9])); if(d[k]==8, e=concat(e, [8])); if(d[k]==9, e=concat(e, [6]))); subst(Pol(e), x, 10) is(n) = my(d=digits(n)); is_palandinv(n) && n!=invert(n) && ispseudoprime(invert(n)) forprime(p=1, 2e8, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Jul 24 2018
1999, for instance, is a prime which rotated upside down through 180 degrees becomes the prime 6661. Hence 1999 is in the sequence.
a(2) = 601 is an invertible prime; 12*601 - 1 = 7211; 12*601 + 1 = 7213; 7211 and 7213 form a twin prime pair. a(4) = 16661 is an invertible prime; 12*16661 - 1 = 199931; 12*16661 + 1 = 199933; 199931 and 199933 form a twin prime pair.
k = 12; Select[lst = {};
fQ[n_] := Block[{allset = {0, 1, 6, 8, 9}, id = IntegerDigits@n}, rid = Reverse[id /. {6 -> 9, 9 -> 6}];Union@Join[id, allset] == allset && PrimeQ@FromDigits@rid && rid != id];Do[If[PrimeQ@n && fQ@n, AppendTo[lst, n]], {n, 12000000}]; lst,
PrimeQ[k# + 1] && PrimeQ[k# - 1] &]
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