A076850 -id:A076850 - cp's OEIS frontend

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

Robert Israel, Feb 11 2025

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

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

Cf. A076850, A096507, A380974.

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;

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

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

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

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

A383971 Triprimes with sum of digits 3.

Original entry on oeis.org

12, 30, 102, 1002, 2001, 10002, 10011, 11001, 20001, 100101, 101001, 110001, 200001, 1000002, 10001001, 10010001, 11000001, 20000001, 100000101, 1000000011, 1000001001, 1000010001, 1000100001, 1001000001, 1010000001, 10000000002, 10000000011, 10000010001, 11000000001, 100000000101, 100000001001

Offset: 1

Robert Israel, May 16 2025

Numbers that are the product of 3 primes, counted with multiplicity, and whose sum of decimal digits is 3.
Since all terms are divisible by 3, the only term ending with 0 is 30. All others are of the form 10^i + 10^j + 1 with 0 <= j <= i.
For each d from 2 to at least 71, there is at least one term with d digits.
Includes 10^k + 2 for k in A076850.
All terms except 12 are squarefree.
All even terms are Zumkeller numbers (A083207). - Ivan N. Ianakiev, May 18 2025

			a(4) = 1002 is a term because 1+0+0+2 = 3 and 1002 = 2 * 3 * 167 is the product of 3 primes, counted with multiplicity.

Robert Israel, Table of n, a(n) for n = 1..380

Intersection of any two of A014612, A050689, and A052217.

Cf. A001222, A007953, A076850, A083207.

istriprime:= proc(n) local F;
  F:= ifactors(n,easy)[2];
  if not hastype(F,symbol) then return convert(F[..,2],`+`)=3 fi;
  F:= remove(hastype,F,symbol);
  if nops(F) > 1 or (nops(F) = 1 and F[1,2] > 1) then return false fi;
  numtheory:-bigomega(n) = 3
end proc:
R:= 12, 30:
for d from 3 to 30 do
  V:= select(istriprime, [seq(seq(10^(d-1) + 10^j + 1,j=0..d-1)]);
  R:= R,op(V);
od:
R;

s={30};imax=11;Do[n=10^i+10^j+1;If[PrimeOmega[n]==3,AppendTo[s,n]],{i,0,imax},{j,0,i}];Sort[s] (* James C. McMahon, Jun 01 2025 *)

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

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