A244444 -id:A244444 - cp's OEIS frontend

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

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

A309835 Numbers k such that k + phi(k) is a repunit.

Original entry on oeis.org

5798663, 5555564201311, 5555574497311, 5555593942711, 66815976110703, 69437045907973255623

Offset: 1

Giovanni Resta, Aug 19 2019

Also in the sequence is 555555555555555555555556288388841217550575591423513701223. - Robert Israel, Aug 20 2019

The number 5975946235638859341313216528710061511 is also in the sequence. - Daniel Suteu, Aug 22 2019

			5798663 is a terms since phi(5798663) = 5312448 and 5798663 + 5312448 = 11111111.

Subsequence of A116018.

Cf. A000010, A002275, A121048, A244444.

isok(k) = my(d=digits(k+eulerphi(k))); vecmin(d)==1 && vecmax(d)==1; \\ Daniel Suteu, Aug 22 2019

a(5) from Daniel Suteu confirmed by Max Alekseyev, Oct 25 2023

a(6) from Max Alekseyev, Nov 30 2023

A291373 a(n) is the smallest number k such that A001065(k) = A002110(n), or 0 if no such k exists.

Original entry on oeis.org

2, 0, 6, 841, 0, 1722, 30018, 0, 0, 0, 4057230930, 0, 0, 92568222856376123089883329681

Offset: 0

Altug Alkan, Aug 23 2017

For n in A057704, 0 < a(n) <= (A002110(n)-1)^2. - Max Alekseyev, Sep 01 2025

			a(5) = 1722 because sigma(1722) - 1722 = 2*3*5*7*11 = A002110(5) and 1722 is the least number with this property.

Cf. A001065, A002110, A057704, A070015, A116017, A153076, A235710, A244444, A279359, A291356, A378099.

a(n) = A070015(A002110(n)). - Michel Marcus, Aug 25 2017

a(7) and a(10) from Giovanni Resta, Aug 23 2017

a(8)-a(9), a(11)-a(13) from Max Alekseyev, Sep 04 2025

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

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A309835 Numbers k such that k + phi(k) is a repunit.

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A291373 a(n) is the smallest number k such that A001065(k) = A002110(n), or 0 if no such k exists.

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