A034054 -id:A034054 - cp's OEIS frontend

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

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

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)

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

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.

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

Cf. A034054.
Cf. A004022, A107612, A107689, A107690, A107691, A107692, A107694, A107695, A107696, A107697, A107698.
Cf. A346274.

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

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 *)

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

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

A034056 Numbers with multiplicative digital root value 9.

Original entry on oeis.org

9, 19, 33, 91, 119, 133, 191, 313, 331, 911, 1119, 1133, 1191, 1313, 1331, 1911, 3113, 3131, 3311, 9111, 11119, 11133, 11191, 11313, 11331, 11911, 13113, 13131, 13311, 19111, 31113, 31131, 31311, 33111, 91111, 111119, 111133, 111191, 111313, 111331

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers with one 9 or two 3s, and zero or more 1s. - Daniel Forgues, Oct 09 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Multiplicative Digital Root
Index entries for 10-automatic sequences.

Cf. A031347.

Cf. A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 0-9).

Module[{nn=6,ne,te},ne=Union[FromDigits/@Flatten[Permutations/@Table[PadRight[{9},n,1],{n,nn}],1]];te=Rest[Union[FromDigits/@ Flatten[ Permutations/@Table[PadRight[{3,3},n,1],{n,nn}],1]]];Join[ne,te]]//Sort (* Harvey P. Dale, Apr 14 2025 *)

There are n(n+1)(n+2)/6 elements up to 10^n, so a(n) is about 10^sqrt(6n).

A276039 Numbers using only digits 1 and 7.

Original entry on oeis.org

1, 7, 11, 17, 71, 77, 111, 117, 171, 177, 711, 717, 771, 777, 1111, 1117, 1171, 1177, 1711, 1717, 1771, 1777, 7111, 7117, 7171, 7177, 7711, 7717, 7771, 7777, 11111, 11117, 11171, 11177, 11711, 11717, 11771, 11777, 17111, 17117, 17171, 17177, 17711, 17717, 17771, 17777

Offset: 1

Vincenzo Librandi, Aug 19 2016

Numbers k such that the product of digits of k is a power of 7.

There are no prime terms whose number of digits is divisible by 3: for every d that is a multiple of 3, every d-digit number j consisting of no digits other than 1's and 7's will have a digit sum divisible by 3, so j will also be divisible by 3. - Mikk Heidemaa, Mar 27 2021

			7717 is in the sequence because 7*7*1*7 = 343 = 7^3.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. similar sequences listed in A276037.

Cf. A020455, A002275, A002281, A034054.

[n: n in [1..24000] | Set(Intseq(n)) subset {1,7}];

Select[Range[20000], IntegerQ[Log[7, Times@@(IntegerDigits[#])]] &] (* or *) Flatten[Table[FromDigits/@Tuples[{1, 7}, n], {n, 6}]]

is(n) = my(d=digits(n), e=[0, 2, 3, 4, 5, 6, 8, 9]); if(#setintersect(Set(d), Set(e))==0, return(1), return(0)) \\ Felix Fröhlich, Aug 19 2016

a(n) = { my(b = binary(n + 1)); b = b[^1]; b = apply(x -> 6*x + 1, b); fromdigits(b) } \\ David A. Corneth, Mar 27 2021

def a(n):
  b = bin(n+1)[3:]
  return int("".join(b.replace("1", "7").replace("0", "1")))
print([a(n) for n in range(1, 47)]) # Michael S. Branicky, Mar 27 2021

def A276039(n): return 6*int(bin(n+1)[3:])+(10**((n+1).bit_length()-1)-1)//9 # Chai Wah Wu, Jun 28 2025

A277061 Numbers with multiplicative digital root > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 53, 57, 61, 62, 63, 64, 66, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 88, 89, 91, 92, 93, 94, 97, 98, 99, 111, 112, 113, 114, 115

Offset: 1

J. Lowell, Sep 26 2016

Question: when will numbers not in this sequence outnumber numbers in this sequence? Up to n = 1249, there are 524 terms, so 525 terms not in this sequence. Up to n = 1522, there are n/2 terms. No n > 1522 has that property. Up to 10^10, only about 1.46% of numbers are a term.

To find how many terms there are up to 10^n, see if A009994(i) is for 2 <= i <= binomial(n + 9, 9). If it is then that adds A047726(A009994(i)) to the total (we don't have to worry about digits 0 in A009994(i) as there aren't any for the specified i). One may put further constraints on i. For example, A009994(i) can't contain an even digit and a 5 in the same number. - David A. Corneth, Sep 27 2016

			25 is not in this sequence because 2*5 = 10 and 1*0 = 0.

Index entries for 10-automatic sequences.

Cf. A009994, A047726.
Cf. A031347, A034048 (complement).
Cf. A028843 (a subsequence).
Union of A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056 (numbers having multiplicative digital roots 1-9).
Cf. A052382 (a supersequence).

Select[Range@ 112, FixedPoint[Times @@ IntegerDigits@ # &, #] > 0 &] (* Michael De Vlieger, Sep 26 2016 *)

is(n) = n=digits(n); while(#n>1,n=digits(prod(i=1,#n,n[i]))); #n>0 \\ David A. Corneth, Sep 27 2016

More terms from Michael De Vlieger, Sep 26 2016

A201021 Composite numbers whose multiplicative digital root is 7.

Original entry on oeis.org

117, 171, 711, 1711, 7111, 11711, 17111, 71111, 111117, 111171, 111711, 117111, 171111, 711111, 1111117, 1111171, 1711111, 7111111, 11111711, 11117111, 11171111, 11711111, 17111111, 111111117, 111111171, 111111711, 111117111, 111171111, 111711111, 117111111, 171111111

Offset: 1

Jaroslav Krizek, Nov 25 2011

Also composite numbers whose product of digits is 7.

Complement of A107693 with respect to A034054. ~

Cf. A107693 (primes whose multiplicative digital root is 7), A034054 (numbers whose multiplicative digital root is 7).

Flatten[Table[Select[FromDigits/@Permutations[PadRight[{7},n,1]],CompositeQ],{n,9}]]//Sort (* Harvey P. Dale, Mar 29 2023 *)

Number 171 is in sequence because 1*7*1=7.

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A107693 Primes with digital product = 7.

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A034056 Numbers with multiplicative digital root value 9.

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A276039 Numbers using only digits 1 and 7.

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A277061 Numbers with multiplicative digital root > 0.

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A201021 Composite numbers whose multiplicative digital root is 7.

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