A096841 -id:A096841 - cp's OEIS frontend

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

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]

A116017 Numbers m such that m + sigma(m) is a repdigit.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 34, 141, 198, 277, 297, 375, 499, 1420, 2651, 2777, 3554, 4999, 19050, 28660, 29128, 49999, 131061, 506311, 3844863, 3852517, 4761903, 4999999, 22222218, 37560831, 133878933, 506767303, 872011214, 1381799253, 1427435733, 2777777777, 3018915632, 3555555554

Offset: 1

Giovanni Resta, Feb 13 2006

From Farideh Firoozbakht, Aug 17 2006: (Start)
(1) If p=(10^(3n+2)-19)/27 is a prime greater than 3 then m=6p is in the sequence because m+sigma(m)=6*(10^(3n+2)-1)/9 (the proof is easy), so m+sigma(m) is a repdigit number. The smallest such terms is 22222218, the next such term is 6*(10^(3*430+2)-1)/9=222...218 which has 1292 digits.
(2) If p=5*10^n-1 is prime then p is in the sequence because p+sigma(p)=10^(n+1)-1, so p+sigma(p) is a repdigit number. 499, 49999, 4999999,... are such terms.
(3) If p=(25*10^(n-1)-7)/9 is prime then p is in the sequence because p+sigma(p)=5*(10^n-1)/9, so p+sigma(p) is a repdigit number. 2, 277, 2777, 2777777777, ... are such terms.
(4) If p=(16*10^(n-1)-7)/9 is prime then m=2p is in the sequence because m+sigma(m)=8*(10^n-1) /9, so m+sigma(m) is a repdigit number. 34, 3554, 3555555554, ... are such terms. (End)

			22222218 + sigma(22222218) = 66666666.

Max Alekseyev, Table of n, a(n) for n = 1..61 (terms 1..45 from Hiroaki Yamanouchi, terms 46..51 from Giovanni Resta)

Contains A244444 as subsequence.

Cf. A116018, A096841.

Do[If[Length[Union[IntegerDigits[n + DivisorSigma[1, n]]]]==1, Print[n]], {n, 60000000}] (* Farideh Firoozbakht, Aug 17 2006 *)

for(n=1, 10^7, d=digits(sigma(n)+n); c=0; for(i=1, #d-1, if(d[i]!=d[i+1], c++; break)); if(c==0, print1(n, ", "))) \\ Derek Orr, Aug 01 2014

from sympy import divisors
A116017 = [n for n in range(1,10**5) if len(set(str(n+sum(divisors(n))))) == 1] # Chai Wah Wu, Aug 11 2014

More terms from Farideh Firoozbakht, Aug 17 2006, Dec 19 2007
a(36)-a(37) from Donovan Johnson, Feb 17 2013
a(38) from Farideh Firoozbakht, Aug 01 2014

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

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

cp's OEIS Frontend

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

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A116017 Numbers m such that m + sigma(m) 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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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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A096843 Primes of form repdigit - 1. Primes whose sum of divisors is a decimal repdigit.

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