A096506 -id:A096506 - cp's OEIS frontend

A096507 Numbers k such that 6*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

1, 2, 6, 8, 9, 11, 20, 23, 41, 63, 66, 119, 122, 149, 252, 284, 305, 592, 746, 875, 1204, 1364, 2240, 2403, 5106, 5776, 5813, 12456, 14235, 39606, 55544, 84239, 275922

Offset: 1

Labos Elemer, Jul 12 2004

nonn

Also numbers k such that (2*10^k + 1)/3 is prime.
These numbers form a near-repdigit sequence (6)w7.
All the terms from k = 2403 through 14235 correspond to primes. - Joao da Silva (zxawyh66(AT)yahoo.com), Oct 03 2005

			k = 9 gives 2000000001/3 = 666666667, which is prime.
k = 20 gives 66666666666666666667, which is prime.

Makoto Kamada, Prime numbers of the form 66...667.
Index entries for primes involving repunits

Cf. A002275, A056657, A093170, A096503, A096504, A096505, A096506, A096508.

Select[Range@ 2500, PrimeQ[FromDigits@ Table[6, {#}] + 1] &] (* or *)
Select[Range@ 2500, PrimeQ[2 (10^# - 1)/3 + 1] &] (* Michael De Vlieger, Jul 04 2016 *)

a(n) = A056657(n) + 1.

More terms from Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004
39606 and 55544 from Serge Batalov, Jun 2009
84239 from Serge Batalov, Jul 06 2009 confirmed as next term by Ray Chandler, Feb 23 2012
a(33) from Kamada data by Tyler Busby, Apr 14 2024

A266141 Number of n-digit primes in which n-1 of the digits are 2's.

Original entry on oeis.org

4, 2, 3, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0

Offset: 1

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

The leading digits must be 2's and only the trailing digit can vary.
For n large a(n) is usually zero.
a(n) <= 4.  If n > 1 and not a multiple of 3, then a(n) <= 2.  It appears that a(n) <= 1 for n > 3. - Chai Wah Wu, Dec 26 2015

			a(3) = 3 since 223, 227 and 229 are all primes.

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1500 terms from Michael De Vlieger and Robert G. Wilson v)

Cf. A265733, A266142, A266143, A266144, A266145, A266146, A266147, A266148, A266149, A084832, A096506, A099409, A099410.

d = 2; Array[Length@ Select[d (10^# - 1)/9 + (Range[0, 9] - d), PrimeQ] &, 100]

use ntheory ":all"; sub a266141 { my $n=shift; return 4 if $n==1; 0+scalar(grep{is_prime("2"x($n-1).$)} 1,3,7,9); } say a266141($) for 1..20; # Dana Jacobsen, Dec 27 2015

from sympy import isprime
def A266141(n):
    return 4 if n==1 else sum(1 for d in '1379' if isprime(int('2'*(n-1)+d))) # Chai Wah Wu, Dec 26 2015

A096846 Numbers n for which 8*R_n - 1 is prime, where R_n = 11...1 is the repunit (A002275) of length n.

Original entry on oeis.org

1, 3, 4, 6, 9, 12, 72, 118, 124, 190, 244, 304, 357, 1422, 2691, 5538, 7581, 21906, 32176, 44358, 120552, 137073, 152260

Offset: 1

Labos Elemer, Jul 15 2004

Also numbers n such that (8*10^n-17)/9 is prime.

The numbers corresponding to a(1)-a(15) are certified prime, the numbers corresponding to a(16)-a(20) are probable primes. a(21) > 10^5. - Robert Price, May 20 2014

			n=6: a(4)=888887 which is prime.

Makoto Kamada, Prime numbers of the form 88...887.
Index entries for primes involving repunits

Cf. A002275, A056695, A093171, A096506, A096507, A096508, A096845.

Do[ If[ PrimeQ[ 8(10^n - 1)/9 - 1], Print[n]], {n, 0, 5000}] (* Robert G. Wilson v, Oct 15 2004; corrected by Derek Orr, Sep 06 2014 *)

for(n=1,10^4,if(ispseudoprime(8*(10^n-1)/9-1),print1(n,", "))) \\ Derek Orr, Sep 06 2014

a(n) = A056695(n) + 1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008
a(18)-a(20) discovered and reported to Makoto Kamada by Erik Branger; added to OEIS by Robert Price, May 20 2014
a(21)-a(23) from Kamada data by Tyler Busby, Apr 23 2024

A069837 Smallest prime which is a concatenation of n primes.

Original entry on oeis.org

2, 23, 223, 2237, 22273, 222323, 2222273, 22222223, 222222227, 2222222377, 22222222223, 222222223273, 2222222222273, 22222222222327, 222222222222227, 2222222222222533, 22222222222223557, 222222222222222577, 2222222222222222327, 22222222222222222253, 222222222222222222277, 2222222222222222222273, 22222222222222222222327

Offset: 1

Amarnath Murthy, Apr 16 2002

Conjecture: For every n there exists an n-digit prime which is composed of the digits 2,3,5 and 7. I.e., no prime > 7 is required in this concatenation. I.e., a(n) of A069637 contains exactly n digits. This is a weaker conjecture than the one by Patrick De Geest in A036937.
If the conjecture is true then this also gives the smallest n-digit prime with prime digits. - Amarnath Murthy, Apr 02 2003
Except for the first term, A096506 lists indices n=2,3,8,11,36,95,101,128,... for which a(n) is of the form 2...23. - M. F. Hasler, Apr 25 2008

Alois P. Heinz, Table of n, a(n) for n = 1..400
C. Caldwell, Prime Curios!: a(101).

Cf. A036937.

Cf. A096506.

f[n_] := Block[{p = 2(10^n - 1)/9}, While[ !PrimeQ[p] || Union[ PrimeQ[ IntegerDigits[p]]] != {True}, p++ ]; p]; Table[ f[n], {n, 1, 20}]

A069837(n)={ local( p=(10^n-1)\9*2-1 ); n=Vec("2357"); until( !setminus( Set(Vec(Str(p))), n), p=nextprime(p+1)); p } /* a more efficient version should check digits one by one and skip to the next possible candidate (i.e., add 12...23 - p%10^d) when a nonprime digit is found */ \\ M. F. Hasler, Apr 25 2008

Edited, corrected and extended by Robert G. Wilson v, Apr 22 2002

A093162 Primes of the form 20*R_k + 3, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

3, 23, 223, 22222223, 22222222223, 222222222222222222222222222222222223

Offset: 1

Rick L. Shepherd, Mar 26 2004

nonn

The next term (a(7)) has 95 digits. - Harvey P. Dale, Dec 26 2022

Makoto Kamada, Prime numbers of the form 22...223.
Index entries for primes involving repunits

Cf. A002275, A056656 (corresponding k, and count of digits 2 in a(n)), A096506.

Select[Table[FromDigits[PadLeft[{3},n,2]],{n,40}],PrimeQ] (* Harvey P. Dale, Dec 26 2022 *)

a(n) = (20*10^A056656(n) + 7)/9 = (2*10^A096506(n) + 7)/9.

Edited by Ray Chandler, Feb 27 2012

A056656 Numbers k such that 20*R_k + 3 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

0, 1, 2, 7, 10, 35, 94, 100, 127, 259, 350, 466, 644, 1010, 1177, 1216, 2441, 3760, 3805, 15616, 26458, 63116, 88544, 93496

Offset: 1

Robert G. Wilson v, Aug 09 2000

nonn

Also numbers k such that (2*10^(k+1)+7)/9 is prime.

Makoto Kamada, Prime numbers of the form 22...223.
Index entries for primes involving repunits

Cf. A093162 (corresponding primes), A096506.

Do[ If[ PrimeQ[ 20*(10^n - 1)/9 + 3 ], Print[n]], {n, 7000}]

a(n) = A096506(n) - 1.

2441 from Rick L. Shepherd, Mar 27 2004
15616 and 26458 from Erik Branger, Jan 31 2010
63116, 88544 and 93496 from Erik Branger, Mar 14 2011; confirmed as next terms by Ray Chandler, Feb 17 2012

cp's OEIS Frontend

A096507 Numbers k such that 6*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A266141 Number of n-digit primes in which n-1 of the digits are 2's.

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A096846 Numbers n for which 8*R_n - 1 is prime, where R_n = 11...1 is the repunit (A002275) of length n.

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A069837 Smallest prime which is a concatenation of n primes.

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A093162 Primes of the form 20*R_k + 3, where R_k is the repunit (A002275) of length k.

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A056656 Numbers k such that 20*R_k + 3 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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Extensions