A096846 -id:A096846 - cp's OEIS frontend

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

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

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

A093171 Primes of the form 80*R_k + 7, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

7, 887, 8887, 888887, 888888887, 888888888887, 888888888888888888888888888888888888888888888888888888888888888888888887

Offset: 1

Rick L. Shepherd, Mar 26 2004

nonn

Primes of the form (8*10^k - 17)/9. - Vincenzo Librandi, Nov 16 2010

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

Cf. A056695 (corresponding k), A096846.

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

A056695 Numbers k such that 80*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

0, 2, 3, 5, 8, 11, 71, 117, 123, 189, 243, 303, 356, 1421, 2690, 5537, 7580, 21905, 32175, 44357

Offset: 1

Robert G. Wilson v, Aug 10 2000

Also numbers k such that (8*10^(k+1)-17)/9 is prime.

a(21) > 10^5. - Robert Price, May 20 2014

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

Cf. A002275, A093171, A096846.

Do[ If[ PrimeQ[80*(10^n - 1)/9 + 7], Print[n]], {n, 0, 5000}]

a(n) = A096846(n) - 1.

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

a(18)-a(20) derived from A096846 by Robert Price, May 20 2014

A096843 Primes of form repdigit - 1. Primes whose sum of divisors is a decimal repdigit.

Original entry on oeis.org

2, 3, 5, 7, 43, 443, 887, 2221, 8887, 444443, 888887, 444444443, 888888887, 444444444443, 888888888887, 222222222222222221, 444444444444444444444444444443, 44444444444444444444444444444443

Offset: 1

Labos Elemer, Jul 15 2004

Union numbers 2, 5 and sequences A093171, A093163 and A091189.

Corresponding values of sigma(a(n)) are in A028987. - Jaroslav Krizek, Mar 19 2013

			n=43: sigma(43)=44;

Giovanni Resta, Table of n, a(n) for n = 1..33 (terms < 10^1000)

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

Missing a(1)=2 and a(3)=5 added by Jaroslav Krizek, Mar 19 2013

A096842 Sigma applied to A096841 produces these repdigits: a[n]=A000203[A096841(n)].

Original entry on oeis.org

1, 3, 4, 7, 6, 8, 44, 222, 444, 888, 444, 888, 888, 2222, 6666, 8888, 8888, 222222, 88888, 222222, 444444, 444444, 888888, 444444, 444444, 666666, 888888, 888888, 888888, 888888, 888888, 444444, 444444, 888888, 888888, 888888, 888888, 888888

Offset: 1

Labos Elemer, Jul 15 2004

			n=43:sigma[43]=44;

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]

cp's OEIS Frontend

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

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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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A093171 Primes of the form 80*R_k + 7, where R_k 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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A056695 Numbers k such that 80*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A096843 Primes of form repdigit - 1. Primes whose sum of divisors is a decimal repdigit.

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A096842 Sigma applied to A096841 produces these repdigits: a[n]=A000203[A096841(n)].

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