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A116018

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

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

(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

LINKS

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

EXAMPLE

5185194 + phi(5185194) = 6666666.

PROG

(Python)

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

(PARI)

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

CROSSREFS

Cf. A116017, A096503, A309835.