A033632 -id:A033632 - cp's OEIS frontend

A226119 Numbers such that sigma(phi(tau(n)))=tau(phi(sigma(n))).

1, 6, 36, 64, 105, 114, 135, 1980, 2016, 3072, 5120, 7056, 7840, 9216, 16320, 18720, 18900, 23100, 23622, 24003, 25536, 26088, 26733, 28455, 29078, 29337, 29700, 29760, 30597, 30894, 30912, 31155, 31496, 31758, 32361, 33782, 34020, 34286, 36000, 36036, 36099

Offset: 1

Paolo P. Lava, May 27 2013

nonn

			29337 is in the sequence since:
sigma(29337)=49152 -> phi(49152)=16384 -> tau(16384)=15.
tau(29337)=16 -> phi(16)=8 -> sigma(8)=15.

Paolo P. Lava and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 270 terms from Paolo P. Lava)

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226117, A226118.

with(numtheory); A226119:=proc(q) local n;
for n from 1 to q do
if sigma(phi(tau(n)))=tau(phi(sigma(n))) then print(n);
fi; od; end: A226119(10^6);

Select[Range[36099], DivisorSigma[1, EulerPhi[DivisorSigma[0, #]]] == DivisorSigma[0, EulerPhi[DivisorSigma[1, #]]] &] (* T. D. Noe, May 28 2013 *)

A227927 Numbers n such that phi(sigma(k))/sigma(phi(k)) < phi(sigma(n))/sigma(phi(n)) for all k < n and n is the smallest positive integer with this property.

Original entry on oeis.org

1, 2, 36, 144, 576, 3600, 14400, 921600, 1040400, 4161600, 8643600, 34574400, 266342400, 700131600, 2800526400, 179233689600, 202338032400, 809352129600

Offset: 1

Vladimir Letsko, Oct 09 2013

nonn

All known terms excluding a(2) are perfect squares.

			36 is in the sequence because phi(sigma(36))/sigma(phi(36)) = 18/7 and for all k < 36 phi(sigma(k))/sigma(phi(k)) < 18/7.

Cf. A227011, A062401, A062402, A033632, A229238.

s:= n -> numtheory:-phi(numtheory:-sigma(n))/numtheory:-sigma(numtheory:-phi(n)):
  a,na,A[1],sA[1]:=1,1,1,1:
1;for i from 2 do ss:=s(i): if ss>a then na:=na+1:A[na]:=ss:a:=ss:sA[na]:=i:print(sA[na]) fi od:

A229238 Numbers k such that phi(sigma(k))/sigma(phi(k)) = 2.

Original entry on oeis.org

2, 4, 16, 18, 64, 100, 450, 1458, 4096, 4624, 28900, 36450, 62500, 65536, 130050, 262144, 281250, 1062882, 1336336, 3334800, 7064400, 8352100, 10156800, 10534050, 18062500, 21193200, 22781250, 26572050, 37584450, 39062500, 48944016, 81281250, 124411716

Offset: 1

Vladimir Letsko, Sep 17 2013

nonn

2^j is in the sequence if and only if 2^{j+1}-1 is a Mersenne prime. In other words 2^j is the "even part" of a perfect number. Thus we have some generalization of perfect numbers.

Odd prime divisors of the first 19 terms of a(n) are exclusively 3, 5, 17, i.e., Fermat's primes, but 3334800 = 2^4*3*5^2*7*397.

			18 is in the sequence because phi(sigma(18)) = phi(39) = 24 = 2*sigma(6) = 2*sigma(phi(18)).

Donovan Johnson, Table of n, a(n) for n = 1..73

Cf. A000010, A000203, A033632.

s:=n->phi(sigma(n))/sigma(phi(n));
for i to 9000000 do if s(i)=2 then print(i) fi od:

isok(n) = (eulerphi(sigma(n)) == 2*sigma(eulerphi(n))); \\ Michel Marcus, Sep 23 2013

Extra term 4624 and more terms from Michel Marcus, Sep 23 2013

A370689 Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

Original entry on oeis.org

1, 1, 3, 1, 7, 3, 3, 7, 1, 7, 9, 7, 14, 3, 15, 1, 31, 1, 39, 5, 7, 3, 9, 15, 7, 7, 39, 7, 7, 5, 9, 31, 21, 31, 15, 7, 91, 39, 5, 31, 15, 7, 24, 7, 5, 3, 9, 31, 8, 7, 21, 10, 49, 39, 15, 15, 91, 7, 45, 31, 28, 9, 91, 1, 31, 7, 9, 7, 21, 5, 6, 5, 65, 91, 3, 91, 21

Offset: 1

Amiram Eldar, Feb 27 2024

			Fractions begin with: 1, 1/2, 3/2, 1/2, 7/2, 3/4, 3, 7/8, 1, 7/6, 9/2, 7/12, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.

Cf. A000010, A000203, A033632, A062401, A062402, A065395, A066930, A289336, A073858 (positions of 1's), A289412, A370690 (denominators).

Cf. A080130, A091724.

Table[DivisorSigma[1, EulerPhi[n]]/EulerPhi[DivisorSigma[1, n]], {n, 1, 100}] // Numerator

a(n) = {my(f = factor(n)); numerator(sigma(eulerphi(f)) / eulerphi(sigma(f)));}

Let f(n) = a(n)/A370690(n) = A062402(n)/A062401(n).
Formulas from De Koninck and Luca (2007):
lim sup_{n->oo} f(n)/log_2(n)^2 = exp(2*gamma) (A091724).
lim inf_{n->oo} f(n)/log_2(n)^2 = delta exists, and exp(-gamma)/40 <= delta <= 2*exp(-gamma).
Sum_{k=1..n} f(k) = c * exp(2*gamma) * log_3(n)^2 * n + O(n * log_3(n)^(3/2)), where c = Product_{p prime} (1 - 3/(p*(p + 1)) + 1/(p^2*(p + 1)) + ((p-1)^3/p^2)*Sum_{k>=3} 1/(p^k-1)) = 0.45782563109026414241... .

A370690 Denominator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 1, 8, 1, 6, 2, 12, 3, 2, 8, 2, 6, 2, 8, 4, 4, 2, 2, 16, 5, 3, 16, 6, 1, 8, 2, 36, 8, 18, 4, 18, 18, 16, 2, 24, 2, 8, 5, 4, 2, 2, 2, 60, 3, 10, 8, 7, 9, 32, 4, 8, 32, 3, 8, 48, 5, 4, 48, 2, 6, 8, 2, 4, 8, 4, 1, 8, 12, 36, 2, 48, 4, 4, 4, 20, 11

Offset: 1

Amiram Eldar, Feb 27 2024

See A370689 for details.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.

Cf. A000010, A000203, A033632, A062401, A062402, A065395, A066930 (positions of 1's), A073858, A289336, A289412, A370689 (numerators).

Table[DivisorSigma[1, EulerPhi[n]]/EulerPhi[DivisorSigma[1, n]], {n, 1, 100}] // Denominator

a(n) = {my(f = factor(n)); denominator(sigma(eulerphi(f)) / eulerphi(sigma(f)));}

A092590 a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.

Original entry on oeis.org

-1, 1, 5, 8, 14, 22, 25, 31, 28, 48, 56, 73, 78, 76, 56, 80, 74, 138, 112, 120, 159, 136, 102, 156, 210, 185, 168, 126, 240, 212, 248, 212, 226, 240, 226, 300, 314, 283, 204, 252, 222, 474, 296, 412, 339, 388, 472, 360, 270, 472, 378, 368, 634, 396, 427, 316, 404, 592, 534, 628, 436, 434, 582, 480, 684, 456, 700, 836

Offset: 1

Labos Elemer, Mar 03 2004

The sequence differs from A065394 since it is not monotonic.

			a(1) = sigma(phi(2))- phi(sigma(2)) = sigma(1)-phi(3) = 1-2 = -1.

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

Cf. A000010, A000040, A000203, A033632, A065393, A065394, A065395, A008331, A008332, A092584, A092585, A092586, A092587, A092588, A092589.

[DivisorSigma(1,EulerPhi(p))-EulerPhi(DivisorSigma(1,p)): p in PrimesUpTo(400)]; // Bruno Berselli, Oct 20 2015

Table[DivisorSigma[1, p-1] - EulerPhi[p+1], {p, Prime[Range[100]]}] (* Amiram Eldar, Jun 09 2024 *)

a(n) = sigma(prime(n)-1) - phi(prime(n)+1) = A008332(n) - A008331(n). - Amiram Eldar, Jun 09 2024

A132793 Numbers n such that sigma(phi(n))-phi(n) = phi(sigma(n)-n).

Original entry on oeis.org

3, 70, 138, 792, 924, 1692, 1932, 2124, 2250, 2988, 3852, 30936, 112644, 189252, 240120, 261660, 263928, 338760, 364308, 379470, 390432, 504216, 529110, 785568, 862290, 917700, 979596, 1022310, 1124220, 1404270, 1434072, 2004372, 2526000

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Aug 31 2007

nonn

Used sigma(n)-n, namely the sum of proper divisors.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000010, A000203, A001229, A018784, A033632, A132794.

with(numtheory); P:=proc(n) local i,j,k; for i from 1 by 1 to n do j:=sigma(phi(i))-phi(i); k:=phi(sigma(i)-i); if j=k then print(i); fi; od; end: P(150000);

Select[Range[2600000],DivisorSigma[1,EulerPhi[#]]-EulerPhi[#]==EulerPhi[ DivisorSigma[1,#]-#]&] (* Harvey P. Dale, Mar 24 2016 *)

isA132793(n)={ if( sigma(eulerphi(n))-eulerphi(n) == eulerphi(sigma(n)-n), 1, 0 ) ; }
{ for(n=2,6000000, if(isA132793(n), print1(n, ", ") ; ) ; ) ; } \\ R. J. Mathar, Nov 11 2007

More terms from R. J. Mathar, Nov 11 2007

Invalid first term removed by Donovan Johnson, Sep 11 2013

A292208 Composite numbers k such that sigma(cototient(k)) = cototient(sigma(k) - k) + cototient(k); that is, f(g(k)) = g(f(k)) where f = A001065 and g = A051953.

Original entry on oeis.org

4, 16, 35, 65, 77, 78, 114, 146, 161, 185, 209, 221, 256, 335, 341, 371, 377, 437, 485, 515, 595, 611, 626, 644, 654, 671, 707, 731, 767, 779, 805, 851, 899, 917, 965, 1007, 1067, 1115, 1157, 1211, 1247, 1271, 1309, 1337, 1385, 1397, 1463, 1495, 1529, 1535, 1577, 1631, 1645, 1691, 1771

Offset: 1

Altug Alkan, Sep 11 2017

nonn

Luca and Pomerance proved that arithmetic functions f(g(n)) and g(f(n)) are independent where f = A001065 and g = A051953. For related details and theorems see Luca & Pomerance link.

			35 = 5*7 is a term because A001065(A051953(35)) = A051953(A001065(35)).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Florian Luca and Carl Pomerance, Local behavior of the composition of the aliquot and co-totient functions, in: G. Andrews and F. Garvan (eds.), Analytic Number Theory, Modular Forms and q-Hypergeometric Series, ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham, 2017; author's copy.

Cf. A000203, A001065, A033632, A051953.

Select[Range@ 1800, Function[n, And[CompositeQ@ n, DivisorSigma[1, n - EulerPhi@ n] == (n - EulerPhi@ n) + # - EulerPhi@ # &[DivisorSigma[1, n] - n]]]] (* Michael De Vlieger, Sep 12 2017 *)

a001065(n) = sigma(n)-n;
a051953(n) = n-eulerphi(n);
lista(nn) = forcomposite(n=4, nn, if(a051953(a001065(n))==a001065(a051953(n)), print1(n, ", ")));

A378315 Odd numbers k such that d(phi(k)) = phi(d(k)), where d=A000005 and phi=A000010.

Original entry on oeis.org

1, 15230439315, 18887708385, 74989937295, 78103226565, 86031400455, 114958521405, 179883837315, 210096608085, 367588711035, 418094581905, 461441147895, 590648954805, 649146021615, 685787041485, 836850895335, 874197762165, 990695282031, 996070731201, 1002913997085, 1016370465201, 1029306324501, 1029869788311, 1039854060045, 1043905592457

Offset: 1

Max Alekseyev, Jan 09 2025

nonn

For n > 1, we have A001222(a(n)) >= 9. The smallest a(n) with A001222(a(n)) = 9 is a(65) = 1244586078645.

Max Alekseyev, Table of n, a(n) for n = 1..65
Tong Lingling et al., Find the smallest odd integer n>1 such that phi(tau(n)) = tau(phi(n)), MathOverflow, 2025.
zhouzixiang2004, DMOPC'21 Contest 10 P5 - Number Theory, DMOJ Monthly Open Programming Competition, 2021.

Subsequence of A078148.

Cf. A000005, A000010, A033632.

A058652 Squarefree n such that sigma(phi(n)) = phi(sigma(n)).

Original entry on oeis.org

1, 29262, 114630, 160986, 179562, 252978, 502878, 528954, 780258, 908070, 1080906, 1826454, 2460786, 2870142, 3934686, 5086722, 5493030, 6001206, 6183078, 6621270, 6668634, 8808234, 9298110, 9752190, 10479282, 11707518, 12263334, 12928254, 13513278

Offset: 1

Robert G. Wilson v, Dec 26 2000

nonn

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A033632.

Select[ Range[ 10^7 ], DivisorSigma[ 1, EulerPhi[ # ] ] == EulerPhi[ DivisorSigma[ 1, # ] ] && Union[ Transpose[ FactorInteger[ # ] ] [ [ 2 ] ] ] == {1} & ]

Prepended missing a(1)=1, Donovan Johnson, Mar 03 2012.

cp's OEIS Frontend

A226119 Numbers such that sigma(phi(tau(n)))=tau(phi(sigma(n))).

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A227927 Numbers n such that phi(sigma(k))/sigma(phi(k)) < phi(sigma(n))/sigma(phi(n)) for all k < n and n is the smallest positive integer with this property.

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A229238 Numbers k such that phi(sigma(k))/sigma(phi(k)) = 2.

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A370689 Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

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A370690 Denominator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

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A092590 a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.

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A132793 Numbers n such that sigma(phi(n))-phi(n) = phi(sigma(n)-n).

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A292208 Composite numbers k such that sigma(cototient(k)) = cototient(sigma(k) - k) + cototient(k); that is, f(g(k)) = g(f(k)) where f = A001065 and g = A051953.

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