A096503 -id:A096503 - cp's OEIS frontend

A096504 Euler-phi applied to A096503 results in these decimal repdigits.

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 4, 6, 8, 8, 6, 8, 22, 8, 8, 22, 66, 44, 88, 44, 88, 66, 44, 88, 88, 222, 88, 88, 222, 444, 444, 888, 888, 444, 888, 888, 888, 888, 888, 888, 444444, 666666, 444444, 888888, 888888, 666666, 888888, 888888, 888888, 888888, 888888

Offset: 1

Labos Elemer, Jul 12 2004

			a(60) = A000010(A096503(60)) = A000010(88888892) = 44444444.
Regular solutions: if p = repdigit + 1 is prime, then phi(p) = repdigit.

Amiram Eldar, Table of n, a(n) for n = 1..182 (calculated using the b-file at A096503)

Cf. A000010, A010785, A096503, A096505.

lista(nn) = {for (n= 1, nn, phin = eulerphi(n); d = digits(e=eulerphi(n)); if (vecmin(d) == vecmax(d), print1(e, ", ")););} \\ Michel Marcus, Sep 07 2014

a(n) = A000010(A096503(n)).

A096508 Numbers k for which 8*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

2, 14, 17, 35, 4175, 4472, 9812, 12260, 12341, 13760, 14576, 53411, 144683, 148328

Offset: 1

Labos Elemer, Jul 12 2004

nonn

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

a(15) > 2*10^5. - Robert Price, Sep 06 2014

			35 is a term because 88888888888888888888888888888888889 (34 8's) is a prime number.

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

Cf. A002275, A056663, A096503-A096507.

select(n -> isprime((8*10^n+1)/9), [$1..10000]); # Robert Israel, Sep 07 2014

Do[ If[ PrimeQ[ 8(10^n - 1)/9 + 1], Print[n]], {n, 0, 30000}] (* Robert G. Wilson v, Oct 15 2004 *)

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

a(n) = A056663(n) + 1.

Four missing terms (9812, 12260, 12341, 13760) added, and a(12)-a(14) added from Kamada data, by Robert Price, Sep 06 2014

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

Original entry on oeis.org

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

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 *)

A116018 Numbers n such that n + phi(n) is a repdigit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 17, 21, 63, 167, 201, 389, 603, 1667, 3795, 3889, 4465, 5926, 50394, 166667, 510042, 2000001, 3888889, 5185194, 5798663, 5925926, 6000003, 32050435, 200000001, 335447667, 365110755, 444766346, 600000003, 1558138862, 1565408702, 1587424430

Offset: 1

Giovanni Resta, Feb 13 2006

(I). If p=(2*10^(3n+1)+7)/27 is prime then m=2p is in the sequence because m+phi(m)=3p-1=2*(10^(3n+1)-1)/9 is a repdigit number. m=2*(2*10^811+7)/27 (a 811-digit number) is the smallest such terms and the next such terms has 4219 digits. - Farideh Firoozbakht, Aug 24 2006
(II). If p=(8*10^(3n+1)+1)/27 is prime then m=2p is in the sequence because m+phi(m)=8*(10^(3n+1)-1)/9 is a repdigit number. 5926 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006
(III). If p=(2*10^n+1)/3 then both numbers 3p & 9p are in the sequence because 3p+phi(3p)=5p-2=3*(10^(n+1)-1)/9 & 9p+ phi(9p)=9*(10^(n+1)-1)/9 are repdigit numbers. 21 & 63 are the smallest such terms. - Farideh Firoozbakht, Aug 24 2006
(IV). All primes p of the form (35*10^n+1)/9 are in the sequence because p+phi(p)=7*(10^n-1)/9 is a repdigit number. 389 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006
(V). All primes p of the form (10^n+2)/6 are in the sequence because p+phi(p)=2p-1=3*(10^n-1)/9 is a repdigit number. 2, 17 & 167 are such terms. - Farideh Firoozbakht, Aug 24 2006, Dec 19 2007

			5185194 + phi(5185194) = 6666666.

Max Alekseyev, Table of n, a(n) for n = 1..98 (terms 1..48 terms from Hiroaki Yamanouchi, terms 49..61 from Giovanni Resta)

Cf. A116017, A096503, A309835.

for(n=1,10^9,d=digits(n+eulerphi(n));if(vecmin(d)==vecmax(d),print1(n,", "))) \\ Derek Orr, Aug 11 2014

from sympy import totient
A116018 = [n for n in range(1,10**6) if len(set(str(n+totient(n)))) == 1] # Chai Wah Wu, Aug 11 2014

More terms from Farideh Firoozbakht, Aug 24 2006

a(35)-a(36) from Donovan Johnson, Feb 19 2013

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

A096505 Primes of the form 1 + repdigit. Primes whose totient is a repdigit.

Original entry on oeis.org

3, 5, 7, 23, 67, 89, 223, 666667, 22222223, 66666667, 666666667, 22222222223, 66666666667, 88888888888889, 88888888888888889, 66666666666666666667, 66666666666666666666667, 88888888888888888888888888888888889, 222222222222222222222222222222222223

Offset: 1

Labos Elemer, Jul 12 2004

			89 is a term since it is a prime, and its totient, 88, is a decimal repdigit.

Amiram Eldar, Table of n, a(n) for n = 1..38

Cf. A000010, A010785, A096503, A096504.

s = {3, 5, 7}; Do[r = (10^k - 1)/9; Do[p = m * r + 1; If[PrimeQ[p], AppendTo[s, p]], {m, {2, 6, 8}}], {k, 2, 50}]; s (* Amiram Eldar, Jun 06 2024 *)

a(19) from Amiram Eldar, Jun 06 2024

A098216 Nonprime numbers m for which cototient(m) is a decimal repdigit.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 21, 25, 27, 30, 35, 49, 60, 86, 90, 93, 120, 121, 145, 159, 172, 175, 195, 231, 245, 253, 279, 291, 327, 365, 497, 535, 559, 693, 703, 737, 886, 979, 981, 993, 1037, 1111, 1121, 1411, 1457, 1517, 1617, 1772, 1774, 2047, 2059

Offset: 1

Labos Elemer, Oct 22 2004

For any odd repdigit k, Goldbach's conjecture implies there are primes p, q with p + q = k + 1, and then (if p <> q) p*q is a term. - Robert Israel, Jul 26 2025

			For m=97002, m-totient(m)=66666.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A096503, A010785.

filter:= proc(n) local t,x;
  t:= n - numtheory:-phi(n);
  if t = 1 then return false fi;
  x:= t mod 10;
  t = x * (10^(ilog10(t)+1)-1)/9
end proc:
select(filter, [$1..3000]); # Robert Israel, Jul 25 2025

ta={{0}};Do[s=Length[Union[IntegerDigits[n-EulerPhi[n]]]]; If[Equal[s, 1]&&!PrimeQ[n], Print[{n, n-EulerPhi[n]}];ta=Append[ta, n]], {n, 1, 100000}];ta=Delete[ta, 1];ta-EulerPhi[ta]

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

cp's OEIS Frontend

A096504 Euler-phi applied to A096503 results in these decimal repdigits.

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A096508 Numbers k for which 8*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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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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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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Mathematica

A116018 Numbers n such that n + phi(n) is a 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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Maple