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

1,1

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

Links

Table of n, a(n) for n=1..17.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A076850, A096507, A380974.

Sequence in context: A134572 A106954 A027755 * A260828 A280651 A089785

Adjacent sequences: A380981 A380982 A380983 * A380985 A380986 A380987

Keyword

nonn,base,more

Author

Robert Israel, Feb 11 2025

Extensions

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

Status

approved