A011774 -id:A011774 - cp's OEIS frontend

A065387 a(n) = sigma(n) + phi(n).

2, 4, 6, 9, 10, 14, 14, 19, 19, 22, 22, 32, 26, 30, 32, 39, 34, 45, 38, 50, 44, 46, 46, 68, 51, 54, 58, 68, 58, 80, 62, 79, 68, 70, 72, 103, 74, 78, 80, 106, 82, 108, 86, 104, 102, 94, 94, 140, 99, 113, 104, 122, 106, 138, 112, 144, 116, 118, 118, 184, 122, 126, 140

Offset: 1

Labos Elemer, Nov 05 2001

a(n) = 2n for n listed in A008578, the prime numbers at the beginning of the 20th century. When a(n) = a(n + 1), n is probably listed in A066198, numbers n where phi changes as fast as sigma (the only exceptions below 10000 are 2 and 854). - Alonso del Arte, Nov 16 2005
A. Makowski proved that n is prime if and only if a(n) = n * d(n), where d is A000005. - Charles R Greathouse IV, Mar 19 2012
If n is semiprime, a(n) = 2n+1+ceiling(sqrt(n))-floor(sqrt(n)). - Wesley Ivan Hurt, May 05 2015
Atanassov proves that a(n) >= n + A001414(n). - Charles R Greathouse IV, Dec 06 2016
a(n) = 2*n+1 iff n is square of prime (A001248), a(n) = 2*(n+1) iff n is squarefree semiprime (A006881). - Bernard Schott, Feb 09 2020

			a(10) = 22 because there are 4 coprimes to 10 below 10, the divisors of 10 add up to 18, and 4 + 18 = 22.

K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe.)
A. Makowski, Aufgaben 339, Elemente der Mathematik 15 (1960), pp. 39-40.

Cf. A000010, A000203, A065388, A015702, A051709, A011774.

See A292768 for partial sums, A051612 for sigma - phi.

[DivisorSigma(1,k)+EulerPhi(k):k in [1..65]]; // Marius A. Burtea, Feb 09 2020

with(numtheory); A065387:=n->phi(n) + sigma(n); seq(A065387(n), n=1..100); # Wesley Ivan Hurt, Apr 08 2014

Table[EulerPhi[n] + DivisorSigma[1,n], {n, 65}] (* Alonso del Arte *)
a[n_] := SeriesCoefficient[Sum[(1+MoebiusMu[k])*x^k/(1-x^k)^2, {k, 1, n}], {x, 0, n}]; Array[a, 63] (* Jean-François Alcover, Sep 29 2017, after Ilya Gutkovskiy *)

a(n) = sigma(n) + eulerphi(n) \\ Harry J. Smith, Oct 17 2009

[sigma(n,1)+euler_phi(n) for n in range(1, 64)] # Stefano Spezia, Jul 20 2025

a(n) = A000203(n) + A000010(n).
a(n) = A051709(n) + 2n. - N. J. A. Sloane, Jun 12 2004
G.f.: Sum_{k>=1} (mu(k) + 1)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Sep 29 2017

A011251 Numbers n such that phi(n) + sigma(n) = 3n.

Original entry on oeis.org

312, 560, 588, 1400, 85632, 147492, 556160, 569328, 1590816, 2013216, 3343776, 4695456, 9745728, 12558912, 22013952, 23336172, 30002960, 52021242, 75007400, 137617728, 153587720, 699117024, 904683264, 2468053248, 2834395104, 21669802880, 48444151296

Offset: 1

N. J. A. Sloane

n is necessarily composite.
From the Math. Rev.: if both q=7*2^{r-2}+2^s-1 and p=49*2^{2r-s-4}+7*2^{r-2}-5*2^{r-s-2}-1 are prime for 1 <= s < r-2, then n=2^r*3*p*q is a solution to the equation phi(n)+sigma(n)=3n . [R. K. Guy]
If 7*2^n-1 is prime then m = 2^(n+2)*3*(7*2^n-1) is in the sequence. Because phi(m) = 2^(n+2)*(7*2^n-2); sigma(m) = 7*2^(n+2)*(2^(n+3)-1) so phi(m)+sigma(m) = 2^(n+2)*((7*2^n-2)+(7*2^(n+3)-7)) = 2^(n+2)*(63*2^n-9) = 3*(2^(n+2)*3*(7*2^n-1)) = 3*m. Hence A112729 = 2^(A001771+2)*3*(7*2^A001771-1) is a subsequence of this sequence. - Farideh Firoozbakht, Dec 01 2005
If both numbers p=7*2^n+2^k-1 & q=49*2^(2n-k)+2^(n-k)*(7*2^k-5)-1 are prime and m=2^(n+2)*3*p*q then phi(m)+sigma(m)=3*m. Namely m is in the sequence. - Farideh Firoozbakht, Jan 11 2007

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B42, p. 151.
David Wells, Prime Numbers: The Most Mysterious Figures in Math, Hoboken, New Jersey, John Wiley & Sons (2005), 75.
Ming Zhi ZHANG, A note on the equation phi(n)+sigma(n)=3n, Sichuan Daxue Xuebao 37 (2000), no. 1, 39-40; MR1755990 (2001a:11009).

Donovan Johnson, Table of n, a(n) for n = 1..35 (terms < 5*10^12)
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Kelley Harris, On the classification of integers n that divide phi(n)+sigma(n), J. Num. Theory 129 (2009) 2093-2110.

Cf. A001771, A112729, A011254, A011774, A015704.

Reap[ Do[ If[ EulerPhi[n] + DivisorSigma[1, n] == 3 n, Print[n]; Sow[n]], {n, 0, 10^8, 2}]][[2, 1]] (* Jean-François Alcover, Feb 16 2012 *)

is(n)=eulerphi(n)+sigma(n)==3*n \\ Charles R Greathouse IV, Nov 27 2013

More terms from Jud McCranie

a(26)-a(27) from Donovan Johnson, Feb 28 2012

A011254 Numbers k such that phi(k) + sigma(k) = 4*k.

Original entry on oeis.org

23760, 59400, 153720, 4563000, 45326160, 113315400, 402831360, 731601000, 803685120, 865950624, 919501200, 1178491680, 3504597120, 3786686400, 6429564000, 14924714400, 25310621952, 26998616736, 53138687040, 86955675840, 513969369984, 1054373308800, 1868445408960

Offset: 1

N. J. A. Sloane

If (sigma(m)-phi(m))/(4*m-sigma(m)-phi(m)) is a prime integer p not dividing m, then p*m is in the sequence. 135230346701100 is in the sequence and not divisible by 24. - Jens Kruse Andersen, Feb 17 2009

If k=80*m is in the sequence and gcd(m,10) = 1 then 200*m is also in the sequence. Proof: phi(200*m) + sigma(200*m) = phi(200)*phi(m) + sigma(200)*sigma(m) = 80*phi(m) + 465*sigma(k) = (5/2)*(32*phi(m) + 186*sigma(m)) = (5/2)*(phi(80)*phi(m) + sigma(80)*sigma(m)) = (5/2)*(phi(80*m) + sigma(80*m)) = (5/2)*(phi(k) + sigma(k)) = (5/2)*(4*k) = 5/2*(4*80*m) = 4*(200*m) so 200*m is in the sequence. - Farideh Firoozbakht, Mar 30 2009

			phi(23760) + sigma(23760) = 5760 + 89280 = 4*23760, so 23760 is in the sequence.

David Wells, Prime Numbers: The Most Mysterious Figures in Math, Hoboken, New Jersey, John Wiley & Sons (2005), p. 75.
Zhang Ming-Zhi (typescript submitted to Unsolved Problems section of Monthly, Oct 01 1996.

Donovan Johnson, Table of n, a(n) for n = 1..25 (terms < 5*10^12)
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Kelley Harris, On the classification of integers n that divide phi(n)+sigma(n), J. Num. Theory 129 (2009) 2093-2110

Cf. A000010, A000203, A011251, A011774, A015704.

Select[Range[1000000], DivisorSigma[1, #] + EulerPhi[#] == 4 # &] (* David Nacin, Feb 28 2012 *)

is(n)=eulerphi(n)+sigma(n)==4*n \\ Charles R Greathouse IV, Nov 27 2013

More terms from Jud McCranie
1178491680 from Farideh Firoozbakht, Jan 31 2006
2 more terms from Jud McCranie, Jan 31 2006
24 divides all known terms of the sequence. If this is true for the next five terms then they are 6429564000, 14924714400, 25310621952, 26998616736 and 53138687040. - Farideh Firoozbakht, Mar 11 2006
More terms from Jens Kruse Andersen, Feb 17 2009
a(21) from Donovan Johnson, Feb 28 2012
a(22)-a(23) from Donovan Johnson, Apr 04 2012

A055681 Numbers k that divide sigma(k)-phi(k).

Original entry on oeis.org

1, 2, 12, 42, 1242, 75960, 1447488, 3506976, 6137440, 10834488, 17156160, 90288000, 431440416, 454460160, 704592000, 1385119360, 1588268480, 10674673152, 24913095480, 31103703540, 56015374080, 80767843200, 129631788000, 463308768000, 469897798656, 834460413696

Offset: 1

Robert G. Wilson v, Jun 09 2000

nonn

a(37) > 10^13. - Giovanni Resta, Jun 12 2013

Giovanni Resta, Table of n, a(n) for n = 1..36 (terms < 10^13)

Cf. A011774, A000203, A000010, A054024, A056075.

with(numtheory): A055681:=n->`if`(sigma(n)-phi(n) mod n=0,n,NULL): seq(A055681(n), n=1..10^5); # Wesley Ivan Hurt, Sep 13 2014

Do[If[Mod[DivisorSigma[1, n]-EulerPhi[n], n]==0, Print[n]], {n, 1, 10^9}]

for(n=1,10^8,if((sigma(n)-eulerphi(n))%n==0,print1(n,", "))) \\ Derek Orr, Sep 13 2014

a(16)-a(26) from Donovan Johnson, Feb 28 2012

A067349 Number of divisors of sigma(n)+phi(n).

Original entry on oeis.org

2, 3, 4, 3, 4, 4, 4, 2, 2, 4, 4, 6, 4, 8, 6, 4, 4, 6, 4, 6, 6, 4, 4, 6, 4, 8, 4, 6, 4, 10, 4, 2, 6, 8, 12, 2, 4, 8, 10, 4, 4, 12, 4, 8, 8, 4, 4, 12, 6, 2, 8, 4, 4, 8, 10, 15, 6, 4, 4, 8, 4, 12, 12, 4, 12, 6, 4, 4, 12, 16, 4, 4, 4, 12, 6, 10, 12, 14, 4, 4, 6, 4, 4, 8, 6, 8, 10, 12, 4, 8, 8, 6, 6, 8

Offset: 1

Labos Elemer, Jan 17 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A011774, A067350, A067351.

Table[ DivisorSigma[ 0, DivisorSigma[ 1, w ]+EulerPhi[ w ] ], {w, 1, 128} ]

a(n) = numdiv(sigma(n)+eulerphi(n)); \\ Michel Marcus, Aug 13 2019

a(n) = A000005(A000010(n) + A000203(n)).

A112729 Numbers of the form 2^(k+2)3(72^k-1) where 72^k-1 is prime.

Original entry on oeis.org

312, 85632, 22013952, 1443107438592, 369435881766912, 24211351590301335552, 103986963299971520879061368832

Offset: 1

Farideh Firoozbakht, Dec 01 2005

This sequence is a subsequence of A011251 and A011774, namely if m is in the sequence then phi(m)+sigma(m)=3*m (see Comments line of A011251).

Number of digits of all the 20 known terms of this sequence are respectively 3, 5, 8, 13, 15, 20, 30, 109, 11069, 13566, 14787, 15722, 20988, 25263, 40594, 42272, 101802, 104453, 107155 and 219110.

			103986963299971520879061368832 is in the sequence because 103986963299971520879061368832=2^(45+2)*3*(7*2^45-1) and 7*2^45-1 is prime.

Cf. A001771. A011251, A011774.

Do[If[PrimeQ[7*2^n-1], Print[3*2^(n+2)*(7*2^n-1)]], {n, 177}]

cp's OEIS Frontend

A065387 a(n) = sigma(n) + phi(n).

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A011251 Numbers n such that phi(n) + sigma(n) = 3n.

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A011254 Numbers k such that phi(k) + sigma(k) = 4*k.

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A055681 Numbers k that divide sigma(k)-phi(k).

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A067349 Number of divisors of sigma(n)+phi(n).

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A112729 Numbers of the form 2^(k+2)*3*(7*2^k-1) where 7*2^k-1 is prime.

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A112729 Numbers of the form 2^(k+2)3(72^k-1) where 72^k-1 is prime.