A382179 Numbers k such that for each digit of k, 2*k*(digit) + 1 is prime.

1, 3, 6, 9, 11, 14, 15, 22, 24, 25, 27, 28, 33, 44, 54, 63, 75, 78, 81, 88, 99, 111, 119, 131, 141, 153, 168, 173, 219, 249, 252, 255, 279, 282, 322, 325, 333, 357, 363, 414, 441, 459, 474, 491, 538, 553, 558, 565, 611, 666, 674, 699, 794, 797, 828, 831, 832, 858, 895, 924, 947, 955

Offset

1,2

Comments

The decision to use the expression 2*k*(digit) instead of k*(digit) is based on the fact that k*(odd) + 1 is never prime. By multiplying k*(digit) by 2, we ensure that any number, regardless of its digits, can appear in the sequence. Additionally, numbers containing the digit 0 are never terms, as 2*k*(0) + 1 is never prime.

When k is restricted to the smallest term with n distinct digits, only 7 terms exist (see A382198).

If the smallest term k is further restricted to a prime number p with n distinct digits, the conditions become significantly more restrictive (see A382127).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..3761 from Jakub Buczak)

Example

63 is a term because both 2*63*6 + 1 and 2*63*3 + 1 are prime.

Maple

filter:= proc(n)
   andmap(t -> isprime(2*n*t+1), convert(n, base, 10))
end proc:
select(filter, [$1..1000]); # Robert Israel, Nov 02 2025

Mathematica

Select[Range[2500], AllTrue[2 * # * IntegerDigits[#] + 1, PrimeQ] &] (* Amiram Eldar, Mar 17 2025 *)

Prog

(PARI) isok(k) = my(d=Set(digits(k))); for (i=1, #d, if (!isprime(2*k*d[i]+1), return(0)); ); return(1); \\ Michel Marcus, Mar 17 2025
(Python)
from sympy import isprime
def ok(n): return all(isprime(2*n*d+1) for d in map(int, set(str(n))))
print([k for k in range(956) if ok(k)]) # Michael S. Branicky, Mar 17 2025

Crossrefs

Cf. A000040, A382127, A382198, A382199. Subset of A052382.

Sequence in context: A310142 A248567 A310143 * A310144 A310145 A223731

Adjacent sequences: A382176 A382177 A382178 * A382180 A382181 A382182

Keyword

nonn,base

Author

Jakub Buczak, Mar 17 2025

Status

approved