A107695 -id:A107695 - cp's OEIS frontend

A107689 Primes with digital product = 3.

3, 13, 31, 113, 131, 311, 11113, 11131, 11311, 113111, 131111, 311111, 11111131, 11111311, 11113111, 11131111, 111111113, 111111131, 111113111, 131111111, 11111111113, 11111111131, 11113111111, 11131111111, 31111111111

Offset: 1

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

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A004022, A107612, A107690, A107691, A107692, A107693, A107694, A107695, A107696, A107697, A107698.

Subsequence of A034050.

Flatten[ Table[ Select[ Sort[ FromDigits /@ Permutations[ Flatten[{3, Table[1, {n}]}]]], PrimeQ[ # ] &], {n, 0, 12}]]

from sympy import isprime
def agen():
  digits = 0
  while True:
    for i in range(digits+1):
      t = int("1"*(digits-i) + "3" + "1"*i)
      if isprime(t): yield t
    digits += 1
g = agen()
print([next(g) for i in range(25)]) # Michael S. Branicky, Mar 13 2021

A107612 Primes with digital product = 2.

Original entry on oeis.org

2, 211, 2111, 111121, 111211, 112111, 1111211, 1111111121, 1111211111, 1121111111, 111111211111, 111211111111, 2111111111111, 111111111111112111, 111111112111111111, 111111211111111111, 112111111111111111

Offset: 1

Zak Seidov, May 17 2005

Corresponding indices of primes in A107611. Cf. A053666, A101987.

Harvey P. Dale, Table of n, a(n) for n = 1..684

Cf. A053666, A101987, A107611.

Cf. A004022, A107689, A107690, A107691, A107692, A107693, A107694, A107695, A107696, A107697, A107698.

for i from 0 to 30 do it:=sum(10^j, j=0..i): for k from 0 to i do if isprime(it+10^k) then printf(`%d,`, it+10^k) fi: od:od: (Sellers)

Flatten[ Table[ Select[ Sort[ FromDigits /@ Permutations[ Flatten[{2, Table[1, {n}]}]]], PrimeQ[ # ] &], {n, 0, 19}]] (* Robert G. Wilson v, May 19 2005 *)
Select[Flatten[Table[FromDigits/@Permutations[PadRight[{2},n,1]],{n,20}]],PrimeQ]//Sort (* Harvey P. Dale, May 28 2017 *)

A107612(n) = prime(A107611(n)).

More terms from Robert G. Wilson v and James Sellers, May 19 2005

A107693 Primes with digital product = 7.

Original entry on oeis.org

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}]

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

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 *)

A107694 Primes with digital product = 8.

Original entry on oeis.org

181, 241, 421, 811, 1181, 1811, 2141, 2221, 2411, 4211, 8111, 21221, 141121, 142111, 411211, 1111181, 1112141, 1121221, 1211141, 1211411, 1212121, 2111411, 2121121, 2211211, 2221111, 2411111, 4121111, 4211111, 11221211, 12111221, 12121121

Offset: 1

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

Cf. A004022, A107612, A107689, A107690, A107691, A107692, A107693, A107695, A107696, A107697, A107698.

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

Union[ Flatten[ Table[ Select[ Sort[ FromDigits /@ Join[ Permutations[ Flatten[{8, Table[1, {n}]}]], Permutations[ Flatten[{2, 4, Table[1, {n - 1}]}]], Permutations[ Flatten[{2, 2, 2, Table[1, {n - 2}]}] ]]], PrimeQ[ # ] & ], {n, 0, 7}]]]
Select[Prime[Range[3 10^6]], Times@@IntegerDigits[#] == 8 &] (* Vincenzo Librandi, Jul 27 2016 *)

A107697 Primes with digital product = 12.

Original entry on oeis.org

43, 223, 431, 1223, 1621, 2161, 2213, 3221, 6121, 6211, 11261, 11621, 12161, 12611, 13411, 21611, 26111, 41113, 41131, 61121, 61211, 111143, 111341, 111431, 112213, 114113, 114311, 121123, 121321, 122131, 123121, 131221, 141131, 141311, 143111

Offset: 1

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

Vincenzo Librandi, Table of n, a(n) for n = 1..86

Cf. A004022, A107612, A107689, A107690, A107691, A107692, A107693, A107694, A107695, A107696, A107698.

[p: p in PrimesUpTo(1000000) | &*Intseq(p) eq 12]; // Vincenzo Librandi, Jul 27 2016

Union[ Flatten[ Table[ Select[ Sort[ FromDigits /@ Join[ Permutations[ Flatten[{2, 6, Table[1, {n - 1}]}]], Permutations[ Flatten[{3, 4, Table[1, {n - 1}]}]], Permutations[ Flatten[{2, 2, 3, Table[1, {n - 2}]}] ]]], PrimeQ[ # ] & ], {n, 0, 5}]]]
Select[Prime[Range[75000]], Times@@IntegerDigits[#] == 12 &] (* Vincenzo Librandi, Jul 27 2016 *)

cp's OEIS Frontend

A107689 Primes with digital product = 3.

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A107612 Primes with digital product = 2.

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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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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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A107696 Primes with digital product = 10.

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