A380984 Primes p such that p*(p-1) consists of exactly two different decimal digits.

5, 7, 11, 17, 67, 101, 167, 1667, 166667, 666667, 66666667, 666666667, 1666666667, 66666666667, 166666666667, 166666666666667, 66666666666666666667

Offset: 1

Author

Robert Israel, Feb 11 2025

Keywords

• nonn
• base
• more

Comments

Primes in A380974.

Contains (10^k+2)/6 for k in A076850 and (2*10^k + 1)/3 for k in A096507. It is conjectured that these sequences are infinite.

The last decimal digit of a(n)*(a(n)-1) is either 0, 2 or 6. - Chai Wah Wu, Feb 19 2025

Examples

			a(5) = 67 is a term because it is prime and 67 * 66 = 4422 consists of digits 2 and 4.
		

Crossrefs

Cf. A076850, A096507, A380974.

Programs

Maple

p:= 1: R:= NULL: count:= 0:
while count < 11 do
p:= nextprime(p);
if nops(convert(convert(p*(p-1),base,10),set)) = 2 then
     R:= R,p; count:= count+1
fi;
od:
R;

Mathematica

Select[Prime[Range[10^6]],Length[Union[IntegerDigits[#(#-1)]]]==2&] (* James C. McMahon, Feb 13 2025 *)

PARI

isok(k) = isprime(k) && #Set(digits(k*(k-1))) == 2; \\ Michel Marcus, Feb 11 2025

Python

from math import isqrt
from itertools import count, combinations, product, islice
from sympy import isprime
def A380984_gen(): # generator of terms
    for n in count(1):
        c = []
        for a in combinations('0123456789',2):
            if '0' in a or '2' in a or '6' in a:
                for b in product(a,repeat=n):
                    if b[0] != '0' and b[-1] in {'0','2','6'} and b != (a[0],)*n and b != (a[1],)*n:
                        m = int(''.join(b))
                        q = isqrt(m)
                        if q*(q+1)==m and isprime(q+1):
                            c.append(q+1)
        yield from sorted(c)
A380984_list = list(islice(A380984_gen(),10)) # Chai Wah Wu, Feb 19 2025

Extensions

a(12)-a(17) from Jinyuan Wang, Feb 12 2025