|
| |
|
|
A340290
|
|
Numbers k that are the representation of primes in base 3 and in base 4.
|
|
2
|
|
|
|
2, 1121, 2021, 2111, 10121, 10211, 11201, 12011, 12121, 12211, 21101, 21211, 22111, 101021, 101111, 110021, 110111, 110221, 111211, 112001, 121001, 121021, 122011, 200111, 201101, 210011, 211021, 211111, 222221, 1000211, 1002011, 1010111, 1011121, 1012201, 1021001
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Except for a(1) = 2, which is the only even prime, all terms end with 1.
The corresponding sequences of primes are A235473 (for base 3) and A235467 (for base 4) (see examples).
As 1381 = 1220011_3 = 111211_4, prime 1381 occurs twice and is the next such prime after 2 (see example), which has a representation in base 3 and a representation in base 4 that are both terms of this sequence.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
a(1) = 2 and 2_3 = 2_4 = 2_10.
a(2) = 1121 because 1121_3 = 43_10 and 1121_4 = 89_10 are primes.
a(3) = 2021 because 2021_3 = 61_10 and 2021_4 = 137_10 are primes.
|
|
|
MATHEMATICA
|
f[n_] := Module[{d = IntegerDigits[n, 3]}, If[PrimeQ[FromDigits[d, 4]], FromDigits[d, 10], 0]]; seq = {}; Do[If[PrimeQ[n], m = f[n]; If[m > 0, AppendTo[seq, m]]], {n, 2, 1000}]; seq (* Amiram Eldar, Jan 03 2021 *)
FromDigits[#]&/@Select[Tuples[{0, 1, 2}, 7], PrimeQ[FromDigits[#, 4]] && PrimeQ[ FromDigits[ #, 3]]&] (* Harvey P. Dale, Dec 15 2021 *)
|
|
|
PROG
|
(PARI) f(n, b) = fromdigits(digits(n, b));
my(vp=primes(700)); setintersect(apply(x->f(x, 3), vp), apply(x->f(x, 4), vp)) \\ Michel Marcus, Jan 04 2021
(PARI) forprime(p=2, 10^3, my(t=digits(p, 3)); if( isprime( fromdigits(t, 4)), print1(fromdigits(t, 10), ", "))) \\ Joerg Arndt, Jan 04 2021
(Python)
from sympy import prime, isprime
from sympy.ntheory.factor_ import digits
A340290_list = [int(s) for s in (''.join(str(d) for d in digits(prime(i), 3)[1:]) for i in range(1, 1000)) if isprime(int(s, 4))] # Chai Wah Wu, Jan 09 2021
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,base
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
