A089675 -id:A089675 - cp's OEIS frontend

A108328 Integers n such that 10^n - 23 is a prime number.

3, 11, 17, 23, 35, 161, 765, 3473, 6887, 27681, 34313

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Jun 30 2005

No additional terms < 40000. - Robert Price, Dec 13 2010
See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "99977".
No other terms < 100,000. - Robert Price, Mar 03 2011

			n = 3 is a member because 10^3 - 23 = 1000 - 23 = 977, which is prime.

Makoto Kamada, List of near-repdigit-related prime numbers.
Index entries for primes involving repunits.

Cf. A089675, A095714, A092767.

lst={};Do[If[PrimeQ[10^n-23], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)

a(9)-a(11) from Robert Price, Dec 13 2010

Edited by Ray Chandler, Dec 23 2010

A108330 Integers k such that 10^k - 29 is a prime number.

Original entry on oeis.org

2, 3, 5, 7, 8, 13, 14, 761, 794, 2216, 3710, 3860, 3937, 5091, 7754, 29091

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Jun 30 2005

The next term, if one exists, is > 100000. - Robert Price, Apr 25 2011

See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "99971".

			k = 8 is a term because 10^8 - 29 = 100000000 - 29 = 99999971, which is prime.

Makoto Kamada, List of near-repdigit-related prime numbers.
Index entries for primes involving repunits.

Cf. A089675, A095714, A092767.

a(14)-a(15) from Sean A. Irvine, Mar 04 2010
a(16) from Robert Price, Dec 15 2010
Edited by Ray Chandler, Dec 23 2010

A069215 Numbers n such that phi(n) = reversal(n).

Original entry on oeis.org

1, 21, 63, 270, 291, 2991, 6102, 46676013, 69460293, 2346534651, 6313047393, 23400000651, 80050617822, 234065340651, 234659934651, 2340000000651, 2530227348360, 2934000006591

Offset: 1

Joseph L. Pe, Apr 11 2002

If 10^n-3 is prime (n is in the sequence A089765) and m=3*(10^n-3) then m is in this sequence, for example 299999999999999991 is a term of this sequence because 299999999999999991=3*(10^17-3) and 17 is in the sequence A089675. So 3*(10^A089675-3) is a subsequence of this sequence, A101700 is this subsequence. - Farideh Firoozbakht, Dec 26 2004
A072395 is a subsequence of this sequence. If m is in the sequence and 10 doesn't divide m then reversal(m) is in the sequence A085331, so see Comments on A085331. - Farideh Firoozbakht, Jan 09 2005
If p=(79*10^(4n+1)-83)/101 is prime then 3p is in the sequence. The proof is easy. 21, 2346534651 & 3*(79*10^2697-83)/101 are the first three such terms. - Farideh Firoozbakht, Apr 22 2008, Aug 16 2008
a(19) > 10^13. - Giovanni Resta, Aug 07 2019

			phi(291) = 192.
phi(6102) = 2016 = reversal(6102), so 6102 belongs to the sequence.

Cf. A101700, A004086, A000010, A085331, A072395, A101700, A102278.

Do[If[EulerPhi[n] == FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 1, 10^5}]

for( n=1,1e9, A004086(n)==eulerphi(n) & print1(n","))

More terms from Farideh Firoozbakht, Aug 31 2004
One more term from Farideh Firoozbakht, Jan 09 2005
a(11)-a(13) from Donovan Johnson, Feb 03 2012
a(14)-a(15) from Giovanni Resta, Oct 28 2012
a(16)-a(18) from Giovanni Resta, Aug 07 2019

A092767 Numbers k such that 10^k - 11 is prime.

Original entry on oeis.org

2, 5, 8, 12, 15, 18, 20, 30, 80, 143, 152, 164, 176, 239, 291, 324, 504, 594, 983, 2894, 22226, 35371, 58437, 67863, 180979

Offset: 1

Carl R. White, Apr 23 2004

Some of the larger terms may only correspond to probable primes.
The numbers corresponding to k = 324, 504, 594 & 983 are certified prime by Primo. - Robert G. Wilson v, Jul 01 2005
a(26) > 2.5*10^5. - Robert Price, Apr 12 2015

			k = 5 is a term because 10^5 - 11 = 100000 - 11 = 99989, which is prime.

Makoto Kamada, Near-repdigit numbers of the form AA...AABA.
Makoto Kamada, Prime numbers of the form 99...9989.
Index entries for primes involving repunits.

Cf. A089675, A086865, A004023, A095714.

Do[ If[ PrimeQ[10^n - 11], Print[n]], {n, 3000}] (* Robert G. Wilson v, Jul 01 2005 *)

for(n=0,5000,if(isprime(10^n-11),print1(n,","))) \\ Ryan Propper, Jun 15 2005

4 more terms from Ryan Propper, Jun 15 2005
Edited by N. J. A. Sloane, May 04 2007
a(21)-a(22) from Robert Price, Dec 12 2010
Edited by Ray Chandler, Dec 23 2010
a(23)=58437 and a(24)=67863 from Robert Price, May 29 2011
a(25) from Kamada data by Robert Price, Apr 12 2015

A266148 Number of n-digit primes in which n-1 of the digits are 9's.

Original entry on oeis.org

4, 6, 7, 7, 8, 10, 7, 13, 8, 8, 11, 13, 8, 11, 13, 14, 10, 9, 7, 11, 9, 13, 10, 19, 5, 10, 14, 7, 10, 9, 9, 15, 13, 8, 7, 9, 10, 11, 10, 13, 5, 12, 15, 7, 12, 7, 12, 11, 13, 11, 8, 13, 13, 13, 12, 12, 9, 9, 15, 14, 9, 8, 13, 11, 15, 17, 10, 8, 11, 10, 6, 16, 8, 8, 8, 15, 9, 11, 14, 7, 10, 11, 16, 17, 11, 10, 12, 16, 8, 15, 7, 11, 11, 10, 7, 12, 6, 10, 8, 9

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

The other digit cannot be 0, 3, 6, or 9, or else the number would not be prime. - N. J. A. Sloane, May 20 2016

			a(3) = 7 since 199, 499, 599, 919, 929, 991 and 997 are all the three-digit primes containing two 9's.

Dana Jacobsen, Table of n, a(n) for n = 1..2500 (first 1215 terms from Michael De Vlieger and Robert G. Wilson v)

Cf. A265733, A266141, A266142, A266143, A266144, A266145, A266146, A266147, A266149, A095714, A089675.

f9[n_] := Block[{cnt = k = 0, r = 9 (10^n - 1)/9, s = Range[0, 9] - 9}, While[k < n, cnt += Length@ Select[r + 10^k * s, PrimeQ@ # && IntegerLength@ # > k &]; k++]; cnt]; Array[f9, 100]

use ntheory ":all"; sub a266148 { my $n = shift; vecsum( map { my $k=$; scalar grep { is_prime("9" x $k . $ . "9" x ($n-$k-1)) } 0+($k>0) .. 8 } 0 .. $n-1 ); } # Dana Jacobsen, Jan 01 2016

from sympy import isprime
def A266148(n):
    return sum(1 for d in range(-9,1) for i in range(n) if isprime(10**n-1+d*10**i)) # Chai Wah Wu, Dec 31 2015

A101700 Numbers of the form 3*(10^n-3), where 10^n-3 is prime.

Original entry on oeis.org

21, 291, 2991, 299999999999999991

Offset: 1

Farideh Firoozbakht, Dec 31 2004

nonn

a(5) = 3*(10^140-3) is 141 digits long and is too large to include.

If m is in this sequence then phi(m)=r(m), so this sequence is a subsequence of A069215. a(n)=3*(10^A089675(n)-3), so a(4)=3*(10^17-3), a(5)=3*(10^140-3), a(6)=3*(10^990-3), a(7)=3*(10^1887-3), a(8)=3*(10^3530-3), a(9)=3*(10^5996-3), a(10)=3*(10^13820-3), a(11)=3*(10^21873-3) & a(12)=3*(10 ^26045-3).

			299999999999999991 is in the sequence because (10^17-3) is prime and 3*(10^17-3)=299999999999999991.

Cf. A086947, A089675, A069215.

Do[If[PrimeQ[10^n-3], Print[3*(10^n-3)]], {n, 150}]
3#&/@Select[10^Range[20]-3,PrimeQ] (* Harvey P. Dale, Mar 23 2022 *)

a(n) = 3*(10^A089675(n) - 3).

A108506 Integers n such that 10^n-59 is prime.

Original entry on oeis.org

2, 3, 4, 8, 20, 38, 95, 248, 263, 303, 304, 410, 438, 548, 688, 1074, 1575, 8364, 9910, 15910, 37344

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Jul 06 2005

Certified primality of numbers corresponding to terms 410, 438, 548, 688, 1074 and 1575 with Primo. - Ryan Propper, Jul 08 2005
No other terms <40000.
See Kamada link - primecount.txt for terms, primesize.txt for discovery details including probable or proved primes - search on "99941".

			8 is a member because: n = 8 gives 10^8-59 = 100000000-59 = 99999941, which is prime.

Makoto Kamada, List of near-repdigit-related prime numbers.
Index entries for primes involving repunits.

Cf. A089675, A095714, A092767, A108327, A108328, A108329, A108330, A108331, A108332, A177418.

a(18)-a(20) from Kamada data by Robert Price, Dec 10 2010
a(21) by Robert Price, Dec 16 2010
Edited by Ray Chandler, Dec 23 2010

A056662 Numbers k such that 90*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

0, 1, 2, 16, 139, 989, 1886, 3529, 5995, 13819, 21872, 26044, 87719, 232598

Offset: 1

Robert G. Wilson v, Aug 09 2000

Also numbers k such that 10^(k+1) - 3 is a prime number.

Makoto Kamada, Prime numbers of the form 99...997.
Index entries for primes involving repunits

Cf. A002275, A089675.

Do[ If[ PrimeQ[ 90*(10^n - 1)/9 + 7 ], Print[n]], {n, 0, 1250}]

a(n) = A089675(n) - 1. - Robert Price, Nov 01 2014

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008

a(13)-a(14) derived from A089675 by Robert Price, Nov 01 2014

A102278 Numbers k such that 78*10^k + 217 is prime.

Original entry on oeis.org

1, 2, 8, 10, 13, 21, 22, 36, 57, 80, 149, 484, 505, 642, 806, 974, 1674, 34177

Offset: 1

Farideh Firoozbakht, Jan 04 2005

If n is a term greater than 2 in this sequence and m = 3*(78*10^n + 217) then phi(m) = reversal(m) (m is in the sequence A069215) because phi(m) = 2*(78*10^n + 216) = 156*10^n + 432 = reversal(234*10^n + 651) = reversal(m).
For example since 8>2 & 8 is in this sequence, for m = 3* (78*10^8 + 217) = 23400000651 phi(m) = reversal(m), so 23400000651 is a term of A069215.
Let f(n,m,r,t) = ((9)(n).78.(0)(m).21.(9)(n))(r).(9)(t).7 where dot between numbers means concatenation and "(m)(n)" means number of m's is n.
In fact I proved that for nonnegative integers n, m, r & t such that r*t = 0 if p = f(n,m,r,t) is prime then phi(3*p) = reversal (3*p). (3*p is in the sequence A069215, some special cases:
Case I, p = f(0,0,0,n-1) = (9)(n-1).7 = 10^n - 3 (see A089675). Case II, p = f(0,n-3,0,0) = 78.(0)(n-3).217 = 78*10^n + 217. Case III, p = f(0,0,n,0) = (7821)(n).7. In this case I found only three such prime p1 = (78217)(0).7 = 7, p2 = (7821)(2).7 = 782178217 & p3 = (7821)(674).7, p3 is a prime with length 2697.
Next term is greater than 8280.
Next term is greater than 24000. - Michael S. Branicky, Mar 22 2023

			8 is in the sequence because 78.(8-3)(0).217 = 7800000217 is prime.

Cf. A089675, A069215, A085331, A101700.

Do[If[PrimeQ[78*10^n + 217], Print[n]], {n, 8280}]

is(n)=ispseudoprime(78*10^n+217) \\ Charles R Greathouse IV, May 22 2017

a(18) from Michael S. Branicky, Oct 15 2024

A108327 Integers n such that 10^n-21 is a prime number.

Original entry on oeis.org

2, 6, 32, 108, 408, 1286, 2268, 2328, 4284, 53558, 181182, 249010

Offset: 1

Julien Peter Benney (jpbenney(AT)ftml.net), Jun 30 2005

a(13) > 2.5*10^5. - Robert Price, Apr 12 2015

			6 is a member because 10^6-21 = 1000000-21 = 999979, which is prime.

Makoto Kamada, Near-repdigit numbers of the form AA...AABA.
Makoto Kamada, Prime numbers of the form 99...9979.
Index entries for primes involving repunits.

Cf. A089675, A095714, A092767.

Edited by Ray Chandler, Dec 23 2010
a(10)=53558 from Robert Price, Mar 03 2011
a(11)-a(12) from Kamada data by Robert Price, Apr 12 2015

cp's OEIS Frontend

A108328 Integers n such that 10^n - 23 is a prime number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A108330 Integers k such that 10^k - 29 is a prime number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A069215 Numbers n such that phi(n) = reversal(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A092767 Numbers k such that 10^k - 11 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A266148 Number of n-digit primes in which n-1 of the digits are 9's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Perl

Python

A101700 Numbers of the form 3*(10^n-3), where 10^n-3 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A108506 Integers n such that 10^n-59 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A056662 Numbers k such that 90*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Views

Author

Keywords

Comments