A004022 -id:A004022 - 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

A107698 Smallest prime whose digital product = n or 0 if impossible.

Original entry on oeis.org

11, 2, 3, 41, 5, 23, 7, 181, 19, 251, 0, 43, 0, 127, 53, 281, 0, 29, 0, 541, 37, 0, 0, 83, 11551, 0, 139, 47, 0, 523, 0, 1481, 0, 0, 157, 149, 0, 0, 0, 12451, 0, 67, 0, 0, 59, 0, 0, 283, 11177, 2551, 0, 0, 0, 239, 0, 1187, 0, 0, 0, 1453, 0, 0, 79, 881, 0, 0, 0, 0, 0, 257, 0

Offset: 1

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

Zeros appear at A068191.

			a(20)=541 because 5*4*1=20 and there is no prime less than a(20) which exhibits this characteristic.

Cf. A004022, A107612, A107689, A107690, A107691, A107692, A107693, A107694, A107695, A107696, A107697.

f[n_] := If[ Max[ First /@ FactorInteger[n]] > 7, 0, p = 1; While[Times @@ IntegerDigits[ Prime[p]] != n, p++ ]; Prime[p]]; Table[ f[n], {n, 30}]

A243534 Numbers n such that the list of all divisors of n contains only 1 distinct digit (in base 10).

Original entry on oeis.org

1, 11, 1111111111111111111, 11111111111111111111111

Offset: 1

Jaroslav Krizek, Jun 13 2014

Union of 1 and A004022 (prime repunits).
The next term has 317 digits.
Numbers n such that A037278(n), A176558(n) and A243360(n) contain only 1 distinct digit.

			11 is in sequence because the list of the divisors of 11: (1, 11) contains only 1 distinct digit.

Cf. A095048, A037278, A176558, A243360.
Sequences of numbers n such that the list of divisors of n contains k distinct digits for 1 <= k <= 10: k = 1: A243534; k = 2: A243535; k = 3: A243536; k = 4: A243537; k = 5: A243538; k = 6: A243539; k = 7: A243540; k = 8: A243541; k = 9: A243542; k = 10: A095050.
Cf. A243543.

[Row n = 1 …10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=1;A(n)); Arrangement of column B]

a(1) = 1; for n >= 2, a(n+1) = A004022 (prime repunits).

A046413 Numbers k such that the repunit of length k (11...11, with k 1's) has exactly 2 prime factors.

Original entry on oeis.org

3, 4, 5, 7, 11, 17, 47, 59, 71, 139, 211, 251, 311, 347, 457, 461

Offset: 1

Patrick De Geest, Jul 15 1998

347, 457, 461 and 701 are also terms. The only other possible terms up to 1000 are 263, 311, 509, 557, 617, 647 and 991; repunits of these lengths are known to be composite but the linked sources do not provide their factors. - Rick L. Shepherd, Mar 11 2003
The Yousuke Koide reference now shows the repunit of length 263 partially factored; 263 is no longer a possible candidate for this sequence. - Ray Chandler, Sep 06 2005
The repunit of length 263 has 3 prime factors; the repunit of length 617 has one known prime factor and a large composite. Possible terms > 1000 are 1117, 1213, 1259, 1291, 1373, 1447, 1607, 1637, 1663, 1669, 1759, 1823, 1949, 1987, 2063 & 2087. - Robert G. Wilson v, Apr 26 2010
All terms are either primes or squares of primes in A004023. In particular, the only composite below a million is 4. - Charles R Greathouse IV, Nov 21 2014
a(17) >= 509. The only confirmed term below 2500 is 701. As of July 2019, no factorization is known for the potential terms 509, 557, 647, 991, 1117, 1259, 1447, 1607, 1637, 1663, 1669, 1759, 1823, 1949, 1987, 2063, 2087, 2111, 2203, 2269, 2309, 2341, 2467, 2503, 2521, ... Unless the least prime factors of the respective composites have fewer than ~80 decimal digits and are thus accessible by massive ECM computations, there is no chance for an extension using current publicly available factorization methods. See links to factordb.com for the status of the factorization of the smallest unknown terms. - Hugo Pfoertner, Jul 30 2019

			7 is a term because 1111111 = 239*4649.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

Patrick De Geest, Repunits prime factors.
Shyam Sunder Gupta, Repunit Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 11, 327-352.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Yousuke Kiode, Factorizations of Repunit Numbers.
Eric Weisstein's World of Mathematics, Repunit.
Status of (10^509-1)/9 in factordb.com.
Status of (10^557-1)/9 in factordb.com.
Status of (10^647-1)/9 in factordb.com.

Cf. A000042, A004022 (repunit primes), A046053, A102782.

Select[Range[60],PrimeOmega[FromDigits[PadRight[{},#,1]]]==2&] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Aug 26 2024 *)

More terms from Rick L. Shepherd, Mar 11 2003

a(13)-a(16) from Robert G. Wilson v, Apr 26 2010

A293663 Circular primes that are not repunits.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939

Offset: 1

Felix Fröhlich, Dec 30 2017

Relative complement of A004022 in A068652.
Conjecture: The sequence is finite.
From Michael De Vlieger, Dec 30 2017: (Start)
Primes > 5 in this sequence must only have digits that are in the reduced residue system modulo 10, i.e., {1, 3, 7, 9}.
There are 54 terms that have 6 or fewer decimal digits, the largest of which is 999331.
a(55) must be larger than 10^11. (End) [Corrected by Felix Fröhlich, Mar 15 + 24 2019]
From Felix Fröhlich, Mar 16 2019: (Start)
a(55) > 10^23 if it exists (cf. De Geest link).
Numbers k such that A262988(k) = A055642(k). (End)

			The numbers resulting from cyclic permutations of the digits of 1193 are 1931, 9311 and 3119, respectively and all those numbers are prime, so 1193, 1931, 3119 and 9311 are terms of the sequence.

Felix Fröhlich, Table of n, a(n) for n = 1..54
P. De Geest, Circular Primes

Cf. A004022, A055642, A068652, A262988, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9).

Select[Prime@ Range[10^5], Function[w, And[AllTrue[Array[FromDigits@ RotateRight[w, #] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits@ # &] (* or *)
Select[Flatten@ Array[FromDigits /@ Most@ Rest@ Tuples[{1, 3, 7, 9}, #] &, 9, 2], Function[w, And[AllTrue[Array[FromDigits@ RotateRight[w, #] &, Length@ w], PrimeQ], Union@ w != {1} ]]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 30 2017 *)

eva(n) = subst(Pol(n), x, 10)
rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
is_circularprime(p) = my(d=digits(p), r=rot(d)); if(vecmin(d)==0, return(0), while(1, if(!ispseudoprime(eva(r)), return(0)); r=rot(r); if(r==d, return(1))))
forprime(p=1, , if(vecmax(digits(p)) > 1, if(is_circularprime(p), print1(p, ", "))))

/* The following is a much faster program that only tests numbers whose decimal expansion consists of digits from the set {1, 3, 7, 9}. */
eva(n) = subst(Pol(n), x, 10)
next_v(vec) = my(k=#vec); if(vecmin(vec)==9, vec=concat(vector(#vec, t, 1), [3]); return(vec)); while(k > 0, if(vec[k]==9, vec[k]=1, if(vec[k]==3, vec[k]=7; return(vec), vec[k]=vec[k]+2, return(vec))); k--)
rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
search(n) = my(d=digits(n), e=[], ed=0); while(1, e=rot(d); while(1, if(!ispseudoprime(eva(e)), break, e=rot(e); if(e==d && ispseudoprime(eva(e)), print1(eva(d), ", "); break))); d=next_v(d))
searchfrom(n) = if(n < 12, forprime(p=n, 10, print1(p, ", ")); search(13), my(d=digits(n)); for(k=1, #d, if(d[k]%2==0, d[k]++, if(d[k]==5, d[k]=7))); search(eva(d)))
/* Start a search from 1 upwards as follows: */
searchfrom(1) \\ Felix Fröhlich, Mar 23 2019

A107690 Primes with digital product = 4.

Original entry on oeis.org

41, 4111, 11411, 12211, 21121, 21211, 22111, 112121, 1114111, 11111141, 11141111, 111112121, 111121121, 112111211, 112112111, 121111121, 121112111, 122111111, 212111111, 1111111411, 1111411111, 11111121121, 11111121211, 11111211121

Offset: 1

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

Vincenzo Librandi, Table of n, a(n) for n = 1..2400

Cf. A004022, A107612, A107689, A107691, A107692, A107693, A107694, A107695, A107696, A107697, A107698.

Subsequence of A199987.

[p: p in PrimesUpTo(10^8) | &*Intseq(p) eq 4]; // Vincenzo Librandi, Jun 30 2017

Flatten[ Table[ Select[ Sort[ FromDigits /@ Join[ Permutations[ Flatten[{4, Table[1, {n}]}]], Permutations[ Flatten[{2, 2, Table[1, {n - 1}]}] ]]], PrimeQ[ # ] &], {n, 0, 10}]]

A107691 Primes with digital product = 5.

Original entry on oeis.org

5, 151, 1151, 1511, 511111, 1111151, 115111111, 1111115111, 1115111111, 1151111111, 111111111511, 111511111111, 1111151111111, 5111111111111, 111111151111111, 111151111111111, 5111111111111111, 111115111111111111111, 1111111111111111111511

Offset: 1

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

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A004022, A107612, A107689, A107690, A107692, A107693, A107694, A107695, A107696, A107697, A107698.

[p: p in PrimesUpTo(3*10^8) | &*Intseq(p) eq 5]; // Vincenzo Librandi, Jul 27 2016

select(isprime,[seq(seq((10^m-1)/9 + 4*10^j,j=0..m-1),m=1..40)]); # Robert Israel, Jan 03 2017

Flatten[ Table[ Select[ Sort[ FromDigits /@ Permutations[Flatten[{5, Table[1, {n}]} ]]], PrimeQ[ # ] &], {n, 0, 21}]]
Select[Prime[Range[3 10^6]], Times@@IntegerDigits[#] == 5 &] (* Vincenzo Librandi, Jul 27 2016 *)

A107692 Primes whose product of digits is 6.

Original entry on oeis.org

23, 61, 1123, 1213, 1231, 1321, 2113, 2131, 2311, 3121, 11161, 11213, 11321, 12113, 13121, 16111, 31121, 111611, 611111, 1111213, 1112113, 1112131, 1131121, 1211311, 2111311, 3112111, 11111161, 11112113, 11211131, 11231111, 11312111

Offset: 1

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

Michael S. Branicky, Table of n, a(n) for n = 1..10199 (all terms with <= 136 digits; terms 1..1000 from Harvey P. Dale)

Cf. A004022, A107612, A107689, A107690, A107691, A107693, A107694, A107695.

Cf. A107696, A107697, A107698.

Union[ Flatten[ Table[ Select[ Sort[ FromDigits /@ Join[ Permutations[ Flatten[{6, Table[1, {n}]}]], Permutations[ Flatten[{2, 3, Table[ 1, {n - 1}]}] ]]], PrimeQ[ # ] &], {n, 0, 7}]]]
Select[Prime[Range[750000]],Times@@IntegerDigits[#]==6&] (* Harvey P. Dale, May 29 2016 *)

from sympy import prod, isprime
from sympy.utilities.iterables import multiset_permutations
def agen(maxdigits):
    for digs in range(1, maxdigits+1):
        for mp in multiset_permutations("1"*(digs-1) + "236", digs):
            if prod(map(int, mp)) == 6:
                t = int("".join(mp))
                if isprime(t): yield t
print(list(agen(8))) # Michael S. Branicky, Jun 16 2021

A107695 Primes with digital product = 9.

Original entry on oeis.org

19, 191, 313, 331, 911, 11119, 111119, 111191, 113131, 131113, 131311, 911111, 1131113, 1131131, 1311131, 1311311, 3111131, 3113111, 11111119, 11111911, 11911111, 111111313, 111111331, 111113113, 111113131, 111131131, 111133111

Offset: 1

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

Harvey P. Dale, Table of n, a(n) for n = 1..9000

Cf. A004022, A107612, A107689, A107690, A107691, A107692, A107693, A107694, A107696, A107697, A107698.

[p: p in PrimesUpTo(3*10^7) | &*Intseq(p) eq 9]; // Vincenzo Librandi, Jul 27 2016

Union[ Flatten[ Table[ Select[ Sort[ FromDigits /@ Join[ Permutations[ Flatten[{9, Table[1, {n}]}]], Permutations[ Flatten[{3, 3, Table[1, {n - 1}]}]]]], PrimeQ[ # ] & ], {n, 0, 8}]]]
Select[Prime[Range[3 10^6]], Times@@IntegerDigits[#] == 9 &] (* Vincenzo Librandi, Jul 27 2016 *)

A107696 Primes with digital product = 10.

Original entry on oeis.org

251, 521, 11251, 12511, 15121, 25111, 111521, 115211, 121151, 151121, 152111, 211151, 511211, 11152111, 11511211, 12111511, 15111211, 15121111, 51111211, 111121151, 111512111, 112111511, 112151111, 112511111, 115211111, 121511111, 151211111

Offset: 1

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

Cf. A004022, A107612, A107689, A107690, A107691, A107692, A107693, A107694, A107695, A107697, A107698.

[p: p in PrimesUpTo(3*10^6) | &*Intseq(p) eq 10]; // Vincenzo Librandi, Jul 27 2016

Flatten[ Table[ Select[ Sort[ FromDigits /@ Permutations[ Flatten[{2, 5, Table[1, {n}]} ]]], PrimeQ[ # ] &], {n, 0, 8}]]
Select[Prime[Range[3 10^6]], Times@@IntegerDigits[#] == 10 &] (* Vincenzo Librandi, Jul 27 2016 *)

cp's OEIS Frontend

A107693 Primes with digital product = 7.

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A107698 Smallest prime whose digital product = n or 0 if impossible.

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A243534 Numbers n such that the list of all divisors of n contains only 1 distinct digit (in base 10).

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A046413 Numbers k such that the repunit of length k (11...11, with k 1's) has exactly 2 prime factors.

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A293663 Circular primes that are not repunits.

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A107690 Primes with digital product = 4.

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A107691 Primes with digital product = 5.

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A107692 Primes whose product of digits is 6.

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