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

1,1

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)

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.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A034054.

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

Cf. A346274.

Sequence in context: A120876 A216073 A086870 * A217717 A122528 A123206

Adjacent sequences: A107690 A107691 A107692 * A107694 A107695 A107696

Keyword

base,nonn

Author

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

Extensions

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

Status

approved