A093177 -id:A093177 - cp's OEIS frontend

A095714 Numbers k such that 9*R_k - 8 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

3, 5, 7, 33, 45, 105, 197, 199, 281, 301, 317, 1107, 1657, 3395, 35925, 37597, 64305, 80139, 221631

Offset: 1

Alonso del Arte, Jul 07 2004

nonn

Also numbers k such that 10^k - 9 is a prime.

			a(2) = 5, since 10^5 - 9 = 99991, which is prime.

Makoto Kamada, Prime numbers of the form 99...991.
Index entries for primes involving repunits

Cf. A002275, A093177, A056696.

Do[ If[ PrimeQ[10^n - 9], Print[n]], {n, 0, 7000}]

a(n) = A056696(n) + 1.

a(12) - a(14) from Robert G. Wilson v, Oct 15 2004
a(15) - a(16) from Jason Earls, Jan 07 2008
a(17) - a(19) from Alexander Gramolin, May 13 2011
Edited by Ray Chandler, Feb 26 2012
Title corrected by Robert Price, Sep 06 2014

A170955 a(n) = 10^n - 9.

Original entry on oeis.org

1, 91, 991, 9991, 99991, 999991, 9999991, 99999991, 999999991, 9999999991, 99999999991, 999999999991, 9999999999991, 99999999999991, 999999999999991, 9999999999999991, 99999999999999991, 999999999999999991, 9999999999999999991, 99999999999999999991, 999999999999999999991

Offset: 1

Vincenzo Librandi, Feb 26 2010

Column 10 of A193871. - Omar E. Pol, Aug 22 2011

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

Cf. A093177, A193871.

[10^n-9: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

CoefficientList[Series[(1 + 80*x)/(1 - 11*x + 10*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
LinearRecurrence[{11,-10},{1,91},20] (* Harvey P. Dale, Aug 20 2015 *)
10^Range[1, 25] - 9 (* Vincenzo Librandi, Jan 03 2016 *)

a(n)=10^n-9 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 10*a(n-1) + 81 for n > 1, a(1) = 1.
G.f.: x*(1+80*x)/((10*x-1)*(x-1)). - R. J. Mathar, Aug 24 2011
From Elmo R. Oliveira, Sep 06 2024: (Start)
E.g.f.: 8 + exp(x)*(exp(9*x) - 9).
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2. (End)

Typo in formula corrected by Jon E. Schoenfield, Jun 19 2010

A056696 Numbers k such that 90*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

2, 4, 6, 32, 44, 104, 196, 198, 280, 300, 316, 1106, 1656, 3394, 35924, 37596, 64304, 80138, 221630

Offset: 1

Robert G. Wilson v, Aug 10 2000

nonn

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

Makoto Kamada, Prime numbers of the form 99...991.
Index entries for primes involving repunits

Cf. A002275, A093177, A095714.

Do[ If[ PrimeQ[90*(10^n - 1)/9 + 1], Print[n]], {n, 0, 7000}]

a(n) = A095714(n) - 1.

a(15) - a(16) from Jason Earls, Jan 07 2008
a(17) - a(19) from Alexander Gramolin, May 13 2011
Edited by Ray Chandler, Feb 26 2012

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.

A092675 Primes of the form 80*R_k + 1, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

881, 8888888888888888881, 8888888888888888888888888888888888888888888888888888888888888888888888888888881

Offset: 1

Rick L. Shepherd, Mar 02 2004

Primes of the form 888...881.
The number of 8's in each term is given by the corresponding term of A056664 and so the first term too large to include above is 888...8881 (with 138 8's).
Primes of the form (8*10^k - 71)/9. - Vincenzo Librandi, Nov 16 2010

Makoto Kamada, Prime numbers of the form 88...881.
Index entries for primes involving repunits

Cf. A056664 (corresponding k).

Cf. A051200, A004022, A093176, A093177, A089346, A093174, A092571.

Select[Table[10 FromDigits[PadRight[{},n,8]]+1,{n,150}],PrimeQ] (* Harvey P. Dale, Aug 07 2019 *)

A105259 Number of distinct prime divisors of 99..91 (with n 9's).

Original entry on oeis.org

0, 2, 1, 2, 1, 3, 1, 4, 2, 3, 3, 3, 3, 4, 3, 4, 3, 6, 2, 4, 4, 3, 3, 5, 4, 7, 4, 6, 2, 6, 3, 6, 1, 2, 3, 5, 3, 10, 4, 7, 5, 4, 6, 7, 1, 7, 2, 6, 3, 5, 5, 6, 4, 6, 2, 8, 4, 7, 3, 5, 4, 11, 2, 7, 5, 8, 6, 5, 5, 7, 2, 8, 4, 7, 5, 6, 4, 6, 5, 9, 3, 9, 4, 7, 2, 9, 4

Offset: 0

Parthasarathy Nambi, Apr 14 2005

			If n=1, then the number of distinct prime divisors of 91 is 2.
If n=2, then the number of distinct prime divisors of 991 is 1 (a prime).
If n=3, then the number of distinct prime divisors of 9991 is 2.

Amiram Eldar, Table of n, a(n) for n = 0..229

Cf. A001221, A093177, A170955.

A105259 := proc(n) local x ;x := [1,seq(9,k=1..n)] ; add(op(i,x)*10^(i-1),i=1..nops(x)) ; numtheory[factorset](%) ; nops(%) ; end proc: # R. J. Mathar, Aug 24 2011

Table[PrimeNu[10^(n + 1) - 9], {n, 0, 50}] (* G. C. Greubel, May 10 2017 *)

a(n) = A001221(A170955(n+1)). - R. J. Mathar, Aug 24 2011

More terms from Amiram Eldar, Jan 24 2020

A109548 Primes of the form aaaa...aa1 where a is 1, 2, 3, 4 or 5.

Original entry on oeis.org

11, 31, 41, 331, 2221, 3331, 4441, 33331, 333331, 3333331, 33333331, 44444444441, 555555555551, 5555555555551, 222222222222222221, 333333333333333331, 1111111111111111111, 11111111111111111111111

Offset: 1

Roger L. Bagula, Jun 26 2005

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..38

Cf. A051200, A004022, A093176, A093177, A089346, A093174, A092571, A089345, A089347.

d[n_] = Mod[n, 6] a = Flatten[Table[Sum[d[k]*10^i, {i, 1, m}] + 1, {m, 1, 50}, {k, 1, 5}]] b = Flatten[Table[If[PrimeQ[a[[i]]] == True, a[[i]], {}], {i, 1, Length[a]}]]
Select[FromDigits/@Flatten[Table[PadLeft[{1},i,#]&/@{1,2,3,4,5},{i,2,80}],1],PrimeQ[#]&] (* Vincenzo Librandi, Dec 12 2011 *)

d=1, 2, 3, 4, 5 a(n) = if prime then Sum[d*10^i, {i, 1, m}] + 1

A109549 Primes of the form aaaa...aa1 where a is 6, 7, 8 or 9.

Original entry on oeis.org

61, 71, 661, 881, 991, 6661, 99991, 9999991, 6666666661, 7777777777771, 666666666666666661, 8888888888888888881, 77777777777777777771, 666666666666666666661, 6666666666666666666661, 77777777777777777777771

Offset: 1

Roger L. Bagula, Jun 26 2005

nonn

Easy-to-remember large primes can be formed in this manner.

Vincenzo Librandi, Table of n, a(n) for n = 1..43

Cf. A051200, A004022, A093176, A093177, A089346, A093174, A092571, A089345, A089347.

d[n_] = If[5 + Mod[n, 6] > 0, 5 + Mod[n, 6], 1] a = Flatten[Table[Sum[d[k]*10^i, {i, 1, m}] + 1, {m, 1, 50}, {k, 1, 4}]] b = Flatten[Table[If[PrimeQ[a[[i]]] == True, a[[i]], {}], {i, 1, Length[a]}]]
Select[FromDigits/@Flatten[Table[PadLeft[{1},i,#]&/@{6,7,8,9},{i,2,100}],1],PrimeQ[#]&] (* Vincenzo Librandi, Dec 12 2011 *)

d=6, 7, 8, 9 a(n) = if prime then Sum[d*10^i, {i, 1, m}] + 1

A109550 Primes of the form aaaa...aa1 where a is 3, 4, 5, 6 or 7.

Original entry on oeis.org

31, 41, 61, 71, 331, 661, 3331, 4441, 6661, 33331, 333331, 3333331, 33333331, 6666666661, 44444444441, 555555555551, 5555555555551, 7777777777771, 333333333333333331, 666666666666666661, 77777777777777777771

Offset: 1

Roger L. Bagula, Jun 26 2005

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..52

Cf. A051200, A004022, A093176, A093177, A089346, A093174, A092571, A089345, A089347.

d[n_] = If[2 + Mod[n, 6] > 0, 2 + Mod[n, 6], 1] a = Flatten[Table[Sum[d[k]*10^i, {i, 1, m}] + 1, {m, 1, 50}, {k, 1, 4}]] b = Flatten[Table[If[PrimeQ[a[[i]]] == True, a[[i]], {}], {i, 1, Length[a]}]]
Select[FromDigits/@Flatten[Table[PadLeft[{1},i,#]&/@{3,4, 5,6,7},{i,2,100}],1],PrimeQ[#]&] (* Vincenzo Librandi, Dec 12 2011 *)

d=3, 4, 5, 6, 7 a(n) = if prime then Sum[d*10^i, {i, 1, m}] + 1

A254648 Numbers n whose square representation in base 10 can be split into three parts whose sum is n.

Original entry on oeis.org

36, 82, 91, 235, 379, 414, 675, 756, 792, 909, 918, 964, 991, 1296, 1702, 1782, 3366, 3646, 3682, 4132, 4906, 5149, 6832, 7543, 8416, 8767, 8856, 9208, 9325, 9586, 9621, 9765, 9901, 9945, 9955, 9991, 12222, 12727, 17271, 22231

Offset: 1

Michel Lagneau, Feb 04 2015

Extension of the Kaprekar numbers (A006886) where the number of parts of n^2 is two. It is probably possible to generalize this property with the division of n^2 into m parts.
By convention, the second and third parts may start with the digit 0, but must be positive. For example, 991 is in the sequence because 991^2 = 982081, which can be split into 982, 08 and 1, and 982 + 08 + 1 = 991. But 100 is not; although 100^2 = 10000 and 100 + 0 + 0 = 100, the second and the third part here are not positive. The number 99 is not in the sequence although 99^2 = 9801 and 98 + 0 + 1 = 99.
Property of the sequence:
The sequence is infinite because the numbers of the form 10^n-9 = 91, 991, 991, ... (A170955) are in the sequence: if m = 99...91 with k digits "9", then m^2 = 99...98200...081 with k-1 digits "9" and k-1 digits "0", and 99...982 + 00...8 + 1 = 99...91 = m.
The prime of the sequence are {379, 9901, ...} union {A093177}.
Calculation method: For each class of squares having k-digit numbers, the number of partitions into 3 parts is n(n+1)/2 (A000217). For instance, if the numbers are of the form (abcde) with k = 5, the 6 partitions into 3 subsets are {a,b,{c,d,e}}, {a,{b,c},{d,e}}, {a,{b,c,d},e}, {{a,b},c,{d,e}}, {{a,b},{c,d},e}, {{a,b,c},d,e} and then we compute the corresponding numbers.
Example: 235^2 = 55225 (abcde) = 55225 => {a,b,{c,d,e}} = {5,5,{2,2,5}} => {5,5,225} and 5+5+225 = 235.

			36^2 = 1296 and 1 + 29 + 6 = 36;
235^2 = 55225 and 5 + 5 + 225 = 235;
1782^2 = 3175524 and 3 + 1755 + 24 = 1782;
12727^2 = 161976529 and 1 + 6197 + 6529 = 12727.

Chai Wah Wu, Table of n, a(n) for n = 1..400

Cf. A000217, A006886, A093177, A170955.

from itertools import combinations
A254648_list, n, n2 = [], 10, 100
while n < 10**4:
    m = str(n2)
    for a in combinations(range(1,len(m)),2):
        x, y, z = int(m[:a[0]]), int(m[a[0]:a[1]]), int(m[a[1]:])
        if y != 0 and z != 0 and x+y+z == n:
            A254648_list.append(n)
            break
    n += 1
    n2 += 2*n-1 # Chai Wah Wu, Aug 27 2017

Removed terms 4879 and 5292 by Chai Wah Wu, Aug 27 2017

cp's OEIS Frontend

A095714 Numbers k such that 9*R_k - 8 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A170955 a(n) = 10^n - 9.

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A056696 Numbers k such that 90*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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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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A092675 Primes of the form 80*R_k + 1, where R_k is the repunit (A002275) of length k.

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A105259 Number of distinct prime divisors of 99..91 (with n 9's).

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Maple

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A109548 Primes of the form aaaa...aa1 where a is 1, 2, 3, 4 or 5.

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A109549 Primes of the form aaaa...aa1 where a is 6, 7, 8 or 9.

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