A116017 -id:A116017 - cp's OEIS frontend

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

Offset: 1

Giovanni Resta, Feb 13 2006

(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

			5185194 + phi(5185194) = 6666666.

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

Cf. A116017, A096503, A309835.

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

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

More terms from Farideh Firoozbakht, Aug 24 2006

a(35)-a(36) from Donovan Johnson, Feb 19 2013

A116001 sigma(n) - phi(n) gives a semiprime (A001358).

Original entry on oeis.org

6, 10, 18, 20, 22, 27, 34, 40, 49, 52, 58, 64, 68, 82, 98, 100, 118, 136, 142, 144, 148, 160, 162, 164, 202, 212, 214, 242, 243, 244, 274, 288, 289, 298, 320, 325, 333, 338, 343, 356, 358, 361, 382, 392, 394, 404, 436, 441, 454, 464, 478, 512, 538, 544, 548

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			sigma(100)-phi(100)=177=3*59

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A116017, A116000.

Select[Range[600],PrimeOmega[DivisorSigma[1,#]-EulerPhi[#]]==2&] (* Harvey P. Dale, Mar 18 2013 *)

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

A244444 Numbers n such that n+sigma(n) is a repunit number.

Original entry on oeis.org

4, 5, 506311, 4761903, 506767303, 5517762660583, 5554746531623, 5555541480743, 5458110152757191

Offset: 1

Farideh Firoozbakht, Aug 01 2014

The numbers 47379454926624737751 and 546139199807860751551844463475591 belong to this sequence. - Giovanni Resta, Aug 17 2019

Also in the sequence is 38808343270779723425176258917550576371890625326889683884600092615. - Daniel Suteu, Aug 23 2019

			sigma(4761903)+4761903 = 11111111.

Subsequence of A116017.

Cf. A309835.

for(n=1, 10^10, d=digits(sigma(n)+n); if(vecmax(d)==1&&vecmin(d)==1, print1(n, ", "))) \\ Derek Orr, Aug 02 2014

from sympy import divisors
[n for n in range(1,10**6) if len(set(str(n+sum(divisors(n))))) == 1 and str(n+sum(divisors(n)))[0] == '1'] # Chai Wah Wu, Aug 04 2014

a(5) from Hiroaki Yamanouchi, Aug 26 2014
a(6)-a(8) from Giovanni Resta, Aug 17 2019
a(9) from Daniel Suteu confirmed by Max Alekseyev, May 23 2025

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.

A116028 Numbers k such that k + sigma(k) + sigma(sigma(k)) is a repdigit.

Original entry on oeis.org

1, 2, 15, 25, 119, 157, 2413, 2623, 8415, 14962, 18303, 66078, 1031747, 62826675, 692799137, 759500195, 1144100379

Offset: 1

Giovanni Resta, Feb 13 2006

a(18) > 5*10^9. - Donovan Johnson, Feb 18 2013

			1031747 + sigma(1031747) + sigma(sigma(1031747)) = 5555555.

Cf. A010785, A116017.

repQ[n_]:=Module[{ds=DivisorSigma[1,n]},Count[DigitCount[n+ds+ DivisorSigma[ 1,ds]],0] == 9]; Select[Range[66100],repQ] (* The program generates the first 12 terms of the sequence. To generate more, increase the Range constant, but it will take a long time to run. *) (* Harvey P. Dale, Oct 08 2020 *)

a(14)-a(17) from Donovan Johnson, Feb 18 2013

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

cp's OEIS Frontend

A116018 Numbers n such that n + phi(n) is a repdigit.

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A116001 sigma(n) - phi(n) gives a semiprime (A001358).

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

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A244444 Numbers n such that n+sigma(n) is a repunit number.

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

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A116028 Numbers k such that k + sigma(k) + sigma(sigma(k)) is a repdigit.

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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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