A015702 -id:A015702 - 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

A018894 Numbers k such that sigma(k)/phi(k) sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 1680, 2520, 4620, 9240, 13860, 18480, 27720, 55440, 110880, 120120, 180180, 240240, 360360, 720720, 1441440, 2162160, 3603600, 4084080, 4324320, 6126120, 12252240, 24504480, 36756720, 61261200

Offset: 1

Michel ten Voorde

nonn

Remarkably similar to but ultimately different from A126098. - Jorg Brown and N. J. A. Sloane, Mar 06 2007
Is a(n+1) <= 2*a(n)? Is a(n) divisible by the primorial p# where p is the largest prime divisor of a(n)? Is a(k) divisible by p# for all k > n + 1? (Cf. A002110.) - David A. Corneth, May 22 2016
From Jud McCranie, Nov 28 2017: (Start)
Yes, a(n+1) <= 2*a(n) -- if m is odd, phi(2*m) = phi(m) and sigma(2*m) = 3*sigma(m).
If m is even then phi(2*m) = 2*phi(m) and sigma(2*m) > 2*sigma(m).
So sigma(2*m)/phi(2*m) > sigma(m)/phi(m). (End)
From David A. Corneth, Sep 10 2020: (Start)
Subsequence of A025487.
Let prime(n)# be the product of the first n primes. Then the LCM of the terms <= 10^40 is 89# * 7# * 5# * (3#)^2 * (2#)^4.
We can assume a larger LCM for terms <= 10^60 namely P# * (13#)^3 * (11#) * (5#) * (3#)^2 * (2#)^4. This gives a total of 466 terms <= 10^75 where P is an arbitrary large prime such that P# <= 10^75.
The LCM of these found terms is a proper divisor and for all primes p <= 13 the exponent is less than the assumed prime. Conjecture: These 466 terms are the terms <= 10^75.
For all 240 terms 1 < t <= 10^40 the following holds: there exists a p|t such that t/p is a term. Conjecture: This holds for all terms t > 1.
Using this technique to find terms I get 6522 terms <= 10^1000 and no conflict with terms found above.
See attached file with terms assuming these conjectures. (End)

David A. Corneth, Table of n, a(n) for n = 1..241 (first 79 terms from Jud McCranie)
Jorg Brown, Comparison of records in sigma(n)/phi(n) and A018892
David A. Corneth, Conjectured 6522 terms <= 10^1000

Cf. A000010, A000203, A015702, A020492, A025487, A126098, A002110.

Flatten@ Function[k, FirstPosition[k, #] & /@ Union@ Rest@ FoldList[Max, 0, k]]@ Array[DivisorSigma[1, #]/EulerPhi@ # &, 10^7] (* Michael De Vlieger, May 27 2016, Version 10 *)

lista(nn) = {mse = 0; for (n=1, nn, se = sigma(n)/eulerphi(n); if (se > mse, print1(n, ", "); mse = se););} \\ Michel Marcus, Jul 10 2015

More terms from Jud McCranie, Nov 09 2001

Initial term added by Arkadiusz Wesolowski, Sep 06 2012

A065385 Numbers m at which value of cototient function (A051953) reaches a new record: cototient(m) > cototient(k) for all k < m.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 102, 108, 114, 120, 126, 132, 138, 144, 150, 168, 180, 198, 204, 210, 240, 252, 264, 270, 294, 300, 330, 360, 378, 390, 420, 450, 462, 480, 504, 510, 540, 546, 570, 600, 630, 660, 690, 714

Offset: 1

Labos Elemer, Nov 05 2001

nonn

For totient values prime numbers give similar records.

			a(8) = 30 because for m = 1...29 the cototient values are all smaller than cototient(30) = 22 = A065386(8) and this is the 8th number at which such a record is reached; analogous sequences are A002093, A002182, A015702 or A005250 for functions other than cototient.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A051953, A065386.

With[{s = Array[# - EulerPhi@ # &, 10^3]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, May 16 2018 *)

r=-1; for(n=1,1000,d=n-eulerphi(n); if(r

{ n=0; x=-1; for (m=1, 10^9, c=m - eulerphi(m); if (c > x, x=c; write("b065385.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 17 2009

a=0; s=0; Do[s=n-EulerPhi[n]; If[s>a, a=s; Print[n]], {n, 1, 10000}]

A065386 Successive record values of the cototient function (A051953).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 22, 24, 30, 32, 36, 44, 46, 48, 54, 60, 66, 70, 72, 78, 88, 90, 92, 94, 96, 110, 120, 132, 138, 140, 162, 176, 180, 184, 198, 210, 220, 250, 264, 270, 294, 324, 330, 342, 352, 360, 382, 396, 402, 426, 440, 486, 500, 514, 522, 528, 550, 588

Offset: 1

Labos Elemer, Nov 05 2001

nonn

			a(8)=22 because for m = 1...29 the cototient values are all smaller than cototient(30)=22, where 30=A065385(8) and 22 is the 8th term in the sequence of such local records.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cototient(A065385(n)).
A006093 gives similar records for the totient function. A002093, A002182, A015702, A005250 are analogous sequences for other functions.
a(n) = A051953(A065385(n)).
Cf. A051953, A065386, A000010.

a=0; s=0; Do[s = n-EulerPhi[n]; If[s>a, a=s; Print[s]], {n, 1, 10000}]
(* Second program: *)
With[{s = Array[# - EulerPhi@ # &, 10^3]}, Union@ FoldList[Max, s]] (* Michael De Vlieger, Nov 03 2017 *)

r=-1; for(n=1,1000,d=n-eulerphi(n); if(r

{ n=0; x=-1; for (m=1, 10^9, c=m - eulerphi(m); if (c > x, x=c; write("b065386.txt", n++, " ", c); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 17 2009

A065388 Record values for sigma(m) + phi(m): sum of sigma and totient is larger than for any previous number.

Original entry on oeis.org

2, 4, 6, 9, 10, 14, 19, 22, 32, 39, 45, 50, 68, 80, 103, 106, 108, 140, 144, 184, 219, 248, 258, 284, 316, 392, 451, 528, 594, 624, 672, 808, 816, 915, 948, 955, 1088, 1266, 1440, 1640, 1704, 1824, 1843, 2020, 2031, 2176, 2208, 2610, 3072, 3304, 3512, 3888

Offset: 1

Labos Elemer, Nov 05 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1288 (terms 1..500 from Harry J. Smith)

Cf. A000010, A000203, A065387, A015702.

{ n=r=0; for (m=1, 10^9, x=sigma(m) + eulerphi(m); if (x > r, r=x; write("b065388.txt", n++, " ", x); if (n==500, return)) ) } \\ Harry J. Smith, Oct 17 2009

a(n) = sigma(A015702(n)) + phi(A015702(n));
a(n) = A000203(A015702(n)) + A000010(A015702(n)).
a(n) = A065387(A015702(n)). - Amiram Eldar, Mar 22 2025

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

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A018894 Numbers k such that sigma(k)/phi(k) sets a new record.

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A065385 Numbers m at which value of cototient function (A051953) reaches a new record: cototient(m) > cototient(k) for all k < m.

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A065386 Successive record values of the cototient function (A051953).

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A065388 Record values for sigma(m) + phi(m): sum of sigma and totient is larger than for any previous number.

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