A084832 -id:A084832 - cp's OEIS frontend

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

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

A165402 a(n) = (2*10^n - 11)/9.

Original entry on oeis.org

1, 21, 221, 2221, 22221, 222221, 2222221, 22222221, 222222221, 2222222221, 22222222221, 222222222221, 2222222222221, 22222222222221, 222222222222221, 2222222222222221, 22222222222222221, 222222222222222221, 2222222222222222221, 22222222222222222221, 222222222222222222221

Offset: 1

Ivan Panchenko, Sep 17 2009

a(n) are also n-1 twos followed by a one.

Sum of n-th row of triangle of powers of 10: 1; 10 1 10; 100 10 1 10 100; 1000 100 10 1 10 100 1000; ... - Philippe Deléham, Feb 24 2014

			a(1) = 1;
a(2) = 10 + 1 + 10 = 21;
a(3) = 100 + 10 + 1 + 10 + 100 = 221;
a(4) = 1000 + 100 + 10 + 1 + 10 + 100 + 1000 = 2221; etc. - _Philippe Deléham_, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Markus Tervooren, Factorizations of (2)w1.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A084832 (positions of primes).

Table[FromDigits[PadLeft[{1},n,2]],{n,20}] (* or *) (2*10^Range[20]-11)/9 (* Harvey P. Dale, Aug 10 2011 *)

From Philippe Deléham, Feb 24 2014: (Start)
a(n) = 10*a(n-1) + 11, a(1)=1.
a(n) = 11*a(n-1) - 10*a(n-2), a(1)=1, a(2)=21.
G.f.: x*(1+10*x)/((1-x)*(1-10*x)). (End)
E.g.f.: 1 + exp(x)*(2*exp(9*x) - 11)/9. - Elmo R. Oliveira, Jun 13 2025

A056660 Numbers k such that 20*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

3, 17, 99, 120, 243, 545, 630, 1493, 2565, 8087

Offset: 1

Robert G. Wilson v, Aug 09 2000

nonn

Also numbers k such that (2*10^(k+1)-11)/9 is a prime.

Makoto Kamada, Prime numbers of the form 22...221.
Index entries for primes involving repunits

Cf. A002275, A084832, A091189 (corresponding primes).

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

a(n) = A084832(n) - 1. - Robert Price, Jan 30 2015

a(8) and a(9) from Rick L. Shepherd, Feb 22 2004

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

A084831 Numbers n such that sum of odd divisors and sum of even divisors are both palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 43, 81, 86, 162, 201, 205, 211, 221, 241, 251, 271, 281, 325, 333, 344, 365, 422, 433, 443, 463, 482, 489, 519, 559, 633, 650, 685, 730, 793, 803, 827, 857, 866, 877, 886, 887, 1419, 1505, 1841, 2021, 2111, 2221, 2305, 2441, 2551, 2561, 2611

Offset: 1

Jason Earls, Jun 05 2003

Primes of form 2*10^n + R(n) (A056700) and 2/9*(-1+10^n)-1 (A084832) are members.

			a(11)=162 because sum of even divisors is 242 and sum of odd divisors is 121.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000593, A056700, A084832.

sodQ[n_]:=Module[{dn=Divisors[n],o,e},o=IntegerDigits[Total[Select[ dn,OddQ]]]; e=IntegerDigits[Total[Select[dn,EvenQ]]]; o== Reverse[o] && e==Reverse[e]]; Select[Range[3000],sodQ] (* Harvey P. Dale, Feb 27 2013 *)

A091189 Primes of the form 20*R_k + 1, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

2221, 222222222222222221, 2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222221

Offset: 1

Rick L. Shepherd, Feb 22 2004

Primes of the form 222...221.

The number of 2's in each term is given by the corresponding term of A056660 and so the first term too large to include above is 222...2221 (with 120 2's).

Makoto Kamada, Prime numbers of the form 22...221.
Index entries for primes involving repunits

Cf. A056660 (corresponding k), A084832.

Cf. A051200, A004022, A093176, A093177, A089346.

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

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A165402 a(n) = (2*10^n - 11)/9.

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A056660 Numbers k such that 20*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A084831 Numbers n such that sum of odd divisors and sum of even divisors are both palindromic.

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A091189 Primes of the form 20*R_k + 1, where R_k is the repunit (A002275) of length k.

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