A096508 -id:A096508 - 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

A266147 Number of n-digit primes in which n-1 of the digits are 8's.

Original entry on oeis.org

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

Offset: 1

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

The leading digits must be 8's and only the trailing digit can vary.

For n large a(n) is usually zero.

			a(3) = 3 since 881, 883, and 887 are all primes.

Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1500

Cf. A265733, A266141, A266142, A266143, A266144, A266145, A266146, A266148, A266149, A099421, A099422, A096846, A096508.

d = 8; Array[Length@ Select[d (10^# - 1)/9 + (Range[0, 9] - d), PrimeQ] &, 100]
Join[{4},Table[Count[Table[10FromDigits[PadRight[{},k,8]]+n,{n,{1,3,7,9}}], ?PrimeQ],{k,110}]] (* _Harvey P. Dale, Jun 22 2021 *)

from _future_ import division
from sympy import isprime
def A266147(n):
    return 4 if n==1 else sum(1 for d in [-7,-5,-1,1] if isprime(8*(10**n-1)//9+d)) # Chai Wah Wu, Dec 27 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

A096519 Solutions to A096509[x]=8, the number of prime-powers [including primes] in neighborhood of x with Ceiling[Log[x]] radius, equals 8.

Original entry on oeis.org

284736, 595953, 855723, 855725, 855726, 855727, 1146785, 1146786, 1146787, 1616612, 1616618, 1616624, 1652884, 1654028, 1718708, 1749272, 1954358, 2176624, 2580658, 2580659, 2580660, 2580661, 2580662, 2831672, 2839942

Offset: 1

Labos Elemer, Jul 12 2004

nonn

Cf. A096508-A096518.

A096506 Numbers n for which 2*R_n + 1 is a prime, where R_n = 11...1 is the repunit (A002275) of length n.

Original entry on oeis.org

1, 2, 3, 8, 11, 36, 95, 101, 128, 260, 351, 467, 645, 1011, 1178, 1217, 2442, 3761, 3806, 15617, 26459, 63117, 88545, 93497

Offset: 1

Labos Elemer, Jul 12 2004

nonn

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

Per Kamada link, 181457, 202059, 262874 are also terms, found by Rytis Slatkevicius. - Michael S. Branicky, Sep 13 2024

			n=36: 222222222222222222222222222222222223 is a prime number.

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

Cf. A056656, A093162, A096503, A096504, A096505, A096507, A096508.

Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 1], Print[n]], {n, 7000}] (* Robert G. Wilson v, Oct 14 2004 *)

a(n) = A056656(n) + 1.

a(20)-a(24) from Kamada link by Ray Chandler, Feb 27 2012

A096841 Numbers n such that sum of divisors of these numbers gives a decimal repdigit.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 43, 146, 365, 438, 443, 803, 887, 2221, 4442, 6663, 8887, 87876, 88183, 153837, 250244, 285597, 292860, 296294, 302877, 307674, 344268, 351612, 380718, 403398, 423260, 441821, 444443, 550238, 579038, 584438, 588974, 593163, 600363

Offset: 1

Labos Elemer, Jul 15 2004

			n=43:sigma[43]=44; regular solutions:repdigit-1=prime.

Giovanni Resta, Table of n, a(n) for n = 1..365

Cf. A096503-A096508, A096841-A096846, A000203.

rd[x_] := Length[Union[IntegerDigits[x]]] Do[s = rd[DivisorSigma[1, n]]; s1 = DivisorSigma[1, n]; If[Equal[s, 1], Print[{n, s1}]; ta[[u]] = n; u = u + 1], {n, 1, 1000000}];ta;DivisorSigma[1, ta]
Select[Range[650000],Length[Union[IntegerDigits[DivisorSigma[1,#]]]]==1&] (* Harvey P. Dale, May 11 2019 *)

A084832 Numbers k such that 2*R_k - 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

4, 18, 100, 121, 244, 546, 631, 1494, 2566, 8088, 262603, 282948, 359860

Offset: 1

Jason Earls, Jun 05 2003

Also numbers k such that (2*10^k-11)/9 is prime.
Larger values correspond to strong pseudoprimes.
a(11) > 10^5. - Robert Price, Sep 06 2014

			a(1) = 4 because 2*(10^4-1)/9-1 = 2221 is prime.
a(2) = 18 means that 222222222222222221 is prime.

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

Cf. A084831, A096503-A096508, A096841-A096846, A002275, A056660.

select(t -> isprime(2*(10^t-1)/9-1),[$1..1000]); # Robert Israel, Sep 07 2014

Do[ If[ PrimeQ[2(10^n - 1)/9 - 1], Print[n]], {n, 0, 7000}] (* Robert G. Wilson v, Oct 14 2004; fixed by Derek Orr, Sep 06 2014 *)

for(n=1, 10^4, if(ispseudoprime(2*(10^n-1)/9-1), print1(n,", "))) \\ Derek Orr, Sep 06 2014

from sympy import isprime
def afind(limit):
  n, twoRn = 1, 2
  for n in range(1, limit+1):
    if isprime(twoRn-1): print(n, end=", ")
    twoRn = 10*twoRn + 2
afind(700) # Michael S. Branicky, Apr 18 2021

a(n) = A056660(n) + 1.

a(8) from Labos Elemer, Jul 15 2004
a(10) from Kamada data by Robert Price, Sep 06 2014
a(11)-a(13) from Kamada data by Tyler Busby, Apr 29 2024

A096845 Numbers n for which 4*R_n - 1 is a prime, where R_n = 11...1 is the repunit (A002275) of length n.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 30, 32, 183, 297, 492, 41316

Offset: 1

Labos Elemer, Jul 15 2004

Also numbers n such that (4*10^n-13)/9 is prime.

a(13) > 10^5. - Robert Price, Oct 25 2014

			n=30 means that 444444444444444444444444444443 is prime.

Makoto Kamada, Prime numbers of the form 44...443.
Index entries for primes involving repunits

Cf. A096503-A096508, A096841-A096846, A000203, A056661.

Do[ If[ PrimeQ[ 4(10^n - 1)/9 - 1], Print[n]], {n, 5000}] (* Robert G. Wilson v, Oct 14 2004 *)
Select[Range[500],PrimeQ[FromDigits[PadLeft[{3},#,4]]]&] (* The program generates the first 11 terms of the sequence. *) (* Harvey P. Dale, Feb 10 2022 *)

a(n) = A056661(n) + 1.

a(12) from Robert Price, Oct 25 2014

A056663 Numbers k such that 80*R_k + 9 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

1, 13, 16, 34, 4174, 4471, 9811, 12259, 12340, 13759, 14575, 53410, 144682, 148327

Offset: 1

Robert G. Wilson v, Aug 09 2000

Also numbers k such that (8*10^(k+1) + 1)/9 is prime.

a(15) > 2*10^5. - Robert Price, Nov 01 2014

Makoto Kamada, Prime numbers of the form 88...889.
Index entries for primes involving repunits

Cf. A002275, A096508.

Do[ If[ PrimeQ[ 80*(10^n - 1)/9 + 9 ], Print[n]], {n, 0, 30000}]

a(n) = A096508(n) - 1. - Robert Price, Nov 01 2014

Intermediate missing terms 9811 (from M. Kamada), 12259, 12340, 13759 added by Serge Batalov, Dec 14 2008

a(12)-a(14) derived from A096508 by Robert Price, Nov 01 2014

A096512 Numbers k such that A096509(k) = 1; i.e., the number of prime powers (including primes) in the neighborhood of k with radius ceiling(log(k)) is 1.

Original entry on oeis.org

54, 89, 90, 91, 95, 115, 119, 143, 145, 204, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 218, 219, 220, 296, 297, 298, 299, 301, 302, 303, 304, 320, 321, 322, 323, 325, 326, 327, 328, 329, 330, 390, 409, 410, 411, 471, 475, 476, 477, 478, 479, 531

Offset: 1

Labos Elemer, Jul 12 2004

nonn

Cf. A096508-A096519.

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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A266147 Number of n-digit primes in which n-1 of the digits are 8'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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A096519 Solutions to A096509[x]=8, the number of prime-powers [including primes] in neighborhood of x with Ceiling[Log[x]] radius, equals 8.

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A096506 Numbers n for which 2*R_n + 1 is a prime, where R_n = 11...1 is the repunit (A002275) of length n.

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A096841 Numbers n such that sum of divisors of these numbers gives a decimal repdigit.

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A084832 Numbers k such that 2*R_k - 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A096845 Numbers n for which 4*R_n - 1 is a prime, where R_n = 11...1 is the repunit (A002275) of length n.

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