A177850
Smallest n-digit emirp with only nonprime digits.
Original entry on oeis.org
149, 1009, 10009, 100049, 1000849, 10000169, 100000891, 1000000009, 10000001041, 100000000669, 1000000000091, 10000000001011, 100000000000469, 1000000000004449, 10000000000001101, 100000000000000049
Offset: 3
a(6) = 100049 because all digits {0,1,4,9} are nonprime, and 100049 is prime, and R(100049) = A004086(100049) = 940001 is prime, and there is no smaller 6-digit number meeting these criteria.
-
isA084984 := proc(n) convert(convert(n,base,10),set) ; if % intersect {2,3,5,7} = {} then true; else false; end if; end proc:
A177850 := proc(n) local a; a := 10^(n-1) ; while not (isA006567(a) and isA084984(a)) do a := nextprime(a) ; end do; if a < 10^n then return a ; else return -1 ; end if; end proc:
seq(A177850(n),n=3..40) ; # R. J. Mathar, May 24 2010
A375171
Square array T(n,k), n>0 and k>0, read by antidiagonals in ascending order, giving the smallest n*k-digit number that, if arranged in an n X k matrix, form k-digit reversible prime in each row and n-digit reversible prime in each column, or -1 if no such number exists.
Original entry on oeis.org
2, 37, 37, 337, 1111, 337, 3257, 111331, 113131, 3257, 32233, 13139731, 113101311, 11933371, 32233, 322573, 1111179779, 113101929311, 119310213191, 1119711779, 322573, 3222223, 111111131397, 113101167919739, 1193100990013911, 111971042937997, 111119111337, 3222223
Offset: 1
T(3,2) = 111331 is the smallest 3*2-digit number that if arranged in a 3 X 2 matrix yields in each row and column an reversible prime, i.e.,
11
13
31
-> 11 (1 time), 13 (1 time), 31 (1 time), 113 (1 time), 131 (1 time) are all reversible primes.
Table begins (upper left corner = T(1,1)):
2 37 337 3257 ...
37 1111 113131 11933371 ...
337 111331 113101311 119310213191 ...
3257 13139731 113101929311 1193100990013911 ...
... ... ... ... ...
-
isp(x) = ispseudoprime(x) && ispseudoprime(fromdigits(Vecrev(digits((x)))));
ispd(x) = ispseudoprime(fromdigits(x)) && ispseudoprime(fromdigits(Vecrev(x)));
vp(n) = select(isp, [10^(n-1)..10^n-1]);
isok(val, n, k) = my(d=digits(val), v=vector(k, i, []), j=1); for (i=1, #d, v[j] = concat(v[j], d[i]); j++; if (j>k, j=1);); for (i=1, k, if (!ispd(v[i]), return(0));); return(1);
T(n,k) = my(v = vp(k), nbp = #v, nb = nbp^n); for (i=0, nb-1, my(d=digits(i, nbp)); if (d==[], d=vector(n)); while(#d x+1, d); my(s=""); for (i=1, #d, s = concat(s, Str(v[d[i]]))); my(val = eval(s)); if (isok(val, n, k), return(val));); \\ Michel Marcus, Aug 08 2024
A375261
Smallest n-digit reversible prime with only prime digits.
Original entry on oeis.org
2, 37, 337, 3257, 32233, 322573, 3222223, 32235223, 322222223, 3222222257, 32222232577, 322222232537, 3222222223333, 32222222332733, 322222222237537, 3222222222223373, 32222222222223353, 322222222222225333, 3222222222222222577, 32222222222222225573, 322222222222222233253
Offset: 1
-
PD:= [2,3,5,7]:
g:= proc(n) local L,d,i,x,y;
L:= convert(n,base,4); d:= nops(L);
x:= add(PD[L[i]+1]*10^(i-1),i=1..d);
y:= add(PD[L[-i]+1]*10^(i-1),i=1..d);
if isprime(x) and isprime(y) then return x fi;
end proc:
f:= proc(d) local k,v;
for k from 4^(d-1) do v:= g(k); if v <> NULL then return v fi od
end proc;
f(1):= 2:
map(f, [$1..30]); # Robert Israel, May 11 2025
-
from sympy import isprime
from itertools import product
def a(n):
if n == 1: return 2
for first in "37":
for rest in product("2357", repeat=n-1):
s = first + "".join(rest)
if isprime(t:=int(s)) and isprime(int(s[::-1])):
return t
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Aug 08 2024
Showing 1-3 of 3 results.
Comments