A107693 Primes with digital product = 7.

7, 17, 71, 1117, 1171, 11117, 11171, 1111711, 1117111, 1171111, 11111117, 11111171, 71111111, 1117111111, 1711111111, 17111111111, 1111171111111, 11111111111111171, 11111111171111111, 1111111111111111171, 1111171111111111111, 1111711111111111111

Offset: 1

Author

Zak Seidov and Robert G. Wilson v, May 20 2005

Keywords

• base
• nonn

Comments

Subsequence of A034054. - Michel Marcus, Jul 27 2016

From Bernard Schott, Jul 12 2021: (Start)

This sequence was the subject of the 1st problem, submitted by USSR, during the 31st International Mathematical Olympiad in 1990 at Beijing, but the jury decided not to use it in the competition.

Problem was: Consider the m-digit numbers consisting of one '7' and m-1 '1'. For what values of m are all these numbers prime? (see the reference).

Answer is: only for m = 1 and m = 2, all these m-digit numbers are primes, so, a(1) = 7, then a(2) = 17 and a(3) = 71.

For other results, see A346274. (End)

Examples

			1117 and 1171 are primes, but 1711 = 29 * 59 and 7111 = 13 * 547; hence a(4) = 1117 and a(5) = 1171.
		

References

• Derek Holton, A Second Step to Mathematical Olympiad Problems, Vol. 7, Mathematical Olympiad Series, World Scientific, 2011, & 8.2. USS 1 p. 260 and & 8.14 Solutions pp 284-287.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..1318 (all terms with <= 1000 digits)
• Index to sequences related to Olympiads.

Crossrefs

Cf. A034054.

Cf. A004022, A107612, A107689, A107690, A107691, A107692, A107694, A107695, A107696, A107697, A107698.

Cf. A346274.

Programs

Magma

[p: p in PrimesUpTo(3*10^8) | &*Intseq(p) eq 7]; // Vincenzo Librandi, Jul 27 2016

Mathematica

Flatten[ Table[ Select[ Sort[ FromDigits /@ Permutations[ Flatten[{7, Table[1, {n}]}]]], PrimeQ[ # ] &], {n, 0, 20}]]
Select[Prime[Range[3 10^6]], Times@@IntegerDigits[#] == 7 &] (* Vincenzo Librandi, Jul 27 2016 *)
Sort[Flatten[Table[Select[FromDigits/@Permutations[PadRight[{7},n,1]],PrimeQ],{n,20}]]] (* Harvey P. Dale, Aug 19 2021 *)

Python

from sympy import isprime
def auptod(maxdigits):
    alst = []
    for d in range(1, maxdigits+1):
        if d%3 == 0: continue
        for i in range(d):
            t = int('1'*(d-1-i) + '7' + '1'*i)
            if isprime(t): alst.append(t)
    return alst
print(auptod(20))  # Michael S. Branicky, Jul 12 2021

Extensions

a(21) and beyond from Michael S. Branicky, Jul 12 2021