A107693 -id:A107693

A107689

Primes with digital product = 3.

+10
17

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

(Python)

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

CROSSREFS

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

Subsequence of A034050.

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A107612

Primes with digital product = 2.

+10
16

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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

MAPLE

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)

MATHEMATICA

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 *)

CROSSREFS

Cf. A053666, A101987, A107611.

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

KEYWORD

base,nonn

AUTHOR

Zak Seidov, May 17 2005

EXTENSIONS

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

STATUS

approved

A107698

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

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Zeros appear at A068191.

LINKS

Table of n, a(n) for n=1..71.

EXAMPLE

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

MATHEMATICA

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}]

CROSSREFS

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

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A107690

Primes with digital product = 4.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

Subsequence of A199987.

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A107691

Primes with digital product = 5.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A107692

Primes whose product of digits is 6.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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 *)

PROG

(Python)

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

CROSSREFS

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

Cf. A107696, A107697, A107698.

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A107695

Primes with digital product = 9.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A107696

Primes with digital product = 10.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..27.

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A107694

Primes with digital product = 8.

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..31.

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved

A107697

Primes with digital product = 12.

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

base,nonn

AUTHOR

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

STATUS

approved