A066198 -id:A066198 - 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

A145749 Numbers n such that sigma(n)+phi(n)=sigma(n+1)+phi(n+1).

Original entry on oeis.org

6, 8, 10, 22, 46, 58, 82, 106, 166, 178, 188, 226, 262, 285, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 902, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2013, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578

Offset: 1

Farideh Firoozbakht, Nov 01 2008

If n/2 is an odd prime and n+1 is prime then n is in the sequence, the proof is easy. 8,188,285,902,2013,... are terms of the sequence which they aren't of such form. This sequence is a subsequence of A066198.

If p is an odd Sophie Germain prime then 2*p is in the sequence. There is no term of the sequence which is of the form 2*p where p is prime and p isn't Sophie Germain prime. A244438 gives terms of the sequence which isn't of the form 2*p where p is prime. - Farideh Firoozbakht, Aug 14 2014

			10 is in the sequence because phi(10) + sigma(10) = 4 + 18 = 22 and phi(11) + sigma(11) = 10 + 12 = 22 also.
12 is not in the sequence because phi(12) + sigma(12) = 4 + 28 = 32 but phi(13) + sigma(13) = 12 + 14 = 26.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A065387, A066198, A145748, A005384.

Select[Range[2600],DivisorSigma[1,# ]+EulerPhi[ # ]==DivisorSigma[1,#+1]+EulerPhi[ #+1]&]

for(n=1,10^4, s=eulerphi(n)+sigma(n); if(s==eulerphi(n+1)+sigma(n+1), print1(n,", "))) /* Derek Orr, Aug 14 2014*/

{n: A065387(n)=A065387(n+1)}.

A145748 Numbers n such that phi(n+1)-phi(n)=sigma(n+1)-sigma(n).

Original entry on oeis.org

2, 854, 751358, 1421637, 8775206, 8892195, 16485944, 31845344, 95494035, 277653495, 380438505, 744048855, 1091725394, 1615353002, 2284844925, 2491028745, 6345217034, 8490513014, 12784909335, 14177454885, 15669084375, 17694356295, 17836667354, 24180347115

Offset: 1

Farideh Firoozbakht, Nov 01 2008

nonn

This sequence is a subsequence of A066198.

Cf. A066198, A145749.

de[n_]:=DivisorSigma[1,n]-EulerPhi[n];Do[If[de[n]==de[n+1],Print[n]],{n,50000000}] (* Firoozbakht *)
Select[Range[10^6], (EulerPhi[# + 1] - EulerPhi[#]) == (DivisorSigma[1, # + 1] - DivisorSigma[1, #]) &] (* Alonso del Arte, Feb 08 2012 *)

a(9)-a(16) from Donovan Johnson, Dec 14 2009

a(17)-a(24) from Donovan Johnson, Feb 08 2012

cp's OEIS Frontend

A066762 Duplicate of A066198.

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

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A145749 Numbers n such that sigma(n)+phi(n)=sigma(n+1)+phi(n+1).

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A145748 Numbers n such that phi(n+1)-phi(n)=sigma(n+1)-sigma(n).

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