1,2

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)

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

22222218 + sigma(22222218) = 66666666.

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

(PARI)

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

(Python)

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

Contains A244444 as subsequence.

Cf. A116018, A096841.

Sequence in context: A177064 A092233 A115895 * A182060 A067033 A067034

Adjacent sequences: A116014 A116015 A116016 * A116018 A116019 A116020

nonn,base

Giovanni Resta, Feb 13 2006

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

approved