A116018 -id:A116018 - 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

A116019 Numbers k such that sigma(k) + phi(k) is a repdigit.

Original entry on oeis.org

1, 2, 3, 4, 10, 11, 21, 49, 186, 207, 221, 342, 406, 3324, 4443, 33324, 43375, 222221, 314000, 344032, 389924, 414806, 987652, 2222221, 190476186, 222087442, 222222221, 422720878, 2222222221, 4444444443

Offset: 1

Giovanni Resta, Feb 13 2006

(1). If m=(2*10^n-11)/9 is product of two distinct primes then m is in the sequence because phi(m)+sigma(m)=phi(p*q)+sigma(p*q)=2(p*q+1)=2m+2=4*(10^n-1)/9, so phi(m)+sigma(m) is a repdigit number. 21, 221, 222221, 2222221, 222222221,... are such terms. - Farideh Firoozbakht, Aug 17 2006
(2). If m=(4*10^n-13)/9 is product of two distinct primes then m is in the sequence because phi(m)+sigma(m)=phi(p*q)+sigma(p*q)=2(p*q+1)=2m+2=8*(10^n-1)/9, so phi(m)+sigma(m) is a repdigit number. 4443, 4444444443, 44444444443,... are such terms. - Farideh Firoozbakht, Aug 17 2006
(3). If p=(25*10^(n-1)-7)/9 is an odd prime then m=12*p is in the sequence because phi(m)+sigma(m)=32p+24=8*(10^(n+1)-1)/9 so phi(m)+sigma(m) is a repdigit number. 3324, 33324, 33333333324,... are such terms. - Farideh Firoozbakht, Aug 17 2006
(4). If n is a nonnegative integer and p=(8*10^(3n+2)-17)/27 is prime then m=14*p is in the sequence because phi(m)+sigma(m)=30p+18=8*(10^(3n+3)-1)/9 is a repdigit number. 406, 414806, 414814814814814814814806, ... are such terms of the sequence. - Farideh Firoozbakht, Aug 01 2014

			sigma(314000)+phi(314000)=888888.

Cf. A116017, A116018, A116020.

Do[If[Length[Union[IntegerDigits[EulerPhi[n] + DivisorSigma[1, n]]]]==1, Print[n]], {n, 280000000}] (* Farideh Firoozbakht, Aug 17 2006 *)

for(n=1,10^7,d=digits(sigma(n)+eulerphi(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

3 more terms from Farideh Firoozbakht, Aug 17 2006

a(28)-a(30) from Donovan Johnson, Jan 16 2012

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

A116020 Numbers n such that sigma(n) - phi(n) is a repdigit greater than 2.

Original entry on oeis.org

4, 8, 9, 18, 25, 27, 28, 57, 62, 85, 123, 192, 218, 258, 259, 261, 322, 403, 632, 662, 693, 1127, 2195, 2218, 2321, 2658, 3548, 4577, 4763, 5597, 5603, 5921, 6662, 7421, 7697, 9617, 9683, 10721, 10877, 11537, 12317, 13323, 17243, 18659, 23363, 26483

Offset: 1

Giovanni Resta, Feb 13 2006

For every prime p sigma(p)-phi(p) is 2, so that case is trivial.
(I). If both numbers p=4*10^n+1 & q=(4*10^n-13)/9 are primes then m=p*q is in the sequence because sigma(m)-phi(m)=8*(10^(n+1)-1)/9 is a repdigit number. Conjecture: 123, 17243 & 1772443 are all such terms. - Farideh Firoozbakht, Aug 24 2006
(II). If p=(10^n-7)/3 is prime then m=2p is in the sequence because sigma(m)-phi(m)=2p+4=6*(10^n-1)/9 is a repdigit number. 62 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006
(III). If p=(4*10^n-31)/9 is prime then m=3p is in the sequence because sigma(m)-phi(m)=2p+6=8*(10^n-1)/9 is a repdigit number. 123 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006
(IV). If p=(8*10^n-17)/9 is a prime then both numbers 4p & 46p are in the sequence because sigma(4p)-phi(4p)=5p+9=4*(10^(n+1)-1)/9 & sigma(46p)-phi(46p)=50p+94=4*(10^(n+2)-1)/9 are repdigit numbers. 28 & 322 are the smallest such terms. - Farideh Firoozbakht, Aug 24 2006
(V). If p=(4*10^n-13)/9 is a prime greater than 3 then m=6p is in the sequence because sigma(m)-phi(m)=10p+14=4*(10^(n+1)-1)/9 is a repdigit number. 258 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006
(VI). If p=(8*10^(2n+1)-179)/99 is prime then m=8p is in the sequence because sigma(m)-phi(m)=11p+19=8*(10^(2n+1)-1)/9 is a repdigit number. 632 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006
(VII). If p=(10^(3n+1)-37)/27 is prime then m=12p is in the sequence because sigma(m)-phi(m)=24p+32=8*(10^(3n+1)-1)/9 is a repdigit number. 4444444428 is the smallest such terms. - Farideh Firoozbakht, Aug 24 2006

			sigma(662) - phi(662) = 666.

Cf. A116017, A116018, A116019.

A116027 Numbers k such that k + phi(k) + phi(phi(k)) is a repdigit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 14, 23, 28, 45, 51, 55, 56, 63, 161, 189, 275, 543, 567, 1373, 13499, 13642, 25963, 27284, 54568, 135347, 138587, 304779, 1296346, 2592692, 5185384, 45994399, 338193377, 419192721, 1492937222, 1724137953, 2453038739, 2985874444, 4687039999

Offset: 1

Giovanni Resta, Feb 13 2006

			5185384 + phi(5185384) + phi(phi(5185384)) = 8888888.

Cf. A010785, A116018.

a(32)-a(39) from Donovan Johnson, Feb 18 2013

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

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A116019 Numbers k such that sigma(k) + phi(k) is a repdigit.

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

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A116020 Numbers n such that sigma(n) - phi(n) is a repdigit greater than 2.

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A116027 Numbers k such that k + phi(k) + phi(phi(k)) is a repdigit.

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