A020488 -id:A020488 - cp's OEIS frontend

A064435 Duplicate of A020488.

Original entry on oeis.org

1, 3, 8, 10, 18, 24, 30

Offset: 1

dead

A062516 Numbers k such that 2*tau(k) = phi(k).

Original entry on oeis.org

5, 9, 15, 28, 40, 72, 84, 90, 120

Offset: 1

Jason Earls, Jul 13 2001

Sequence is finite, since for large k and suitable constants and epsilon: phi(k) - 2*tau(k) > c1*k^(2/3) - 4*c2*k^(1/2) > 0 if k > c3, so phi(k) - 2*tau(k) > 0, QED. Moreover, phi(k) = m*tau(k) has at most finitely many solutions for any constant m or even for slowly increasing functions like m(k) = k^(epsilon). - Labos Elemer, Jul 20 2001

Cf. A112954, A020488, A063469, A063470.

Select[Range[150],2*DivisorSigma[0,#]==EulerPhi[#]&] (* Harvey P. Dale, Jun 28 2022 *)

for(n=1,1000000, if(numdiv(n)*2==eulerphi(n),print(n),))

"full" keyword from Max Alekseyev, Mar 01 2010

A112954 Number of numbers m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005.

Original entry on oeis.org

7, 9, 10, 9, 7, 17, 4, 17, 14, 15, 7, 19, 2, 16, 20, 21, 0, 29, 0, 29, 9, 13, 7, 32, 7, 11, 23, 21, 7, 39, 0, 19, 17, 4, 11, 44, 2, 0, 11, 41, 7, 24, 2, 19, 30, 11, 0, 55, 4, 23, 7, 21, 7, 46, 9, 27, 4, 11, 0, 61, 0, 0, 27, 29, 9, 30, 2, 10, 19, 31, 0, 57, 2, 9, 27, 4, 4, 30, 2, 50, 29, 9

Offset: 1

Reinhard Zumkeller, Oct 07 2005

nonn

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A020491, A020488, A062516, A063469, A063470, A112955, A175667.

More terms from Max Alekseyev, Mar 01 2010

A063469 Numbers n such that tau(n)*3 = phi(n).

Original entry on oeis.org

7, 21, 26, 56, 70, 78, 108, 126, 168, 210

Offset: 1

Jason Earls, Jul 26 2001

"phi(n)=k*Tau[n] has at most finitely many solutions for any constant k or even for slowly increasing functions like k(n)=n^(epsilon)." - Labos Elemer, Jul 20 2001

Cf. A062516.

Cf. A112954, A020488, A063470.

Select[Range[300],3DivisorSigma[0,#]==EulerPhi[#]&] (* Harvey P. Dale, Sep 15 2016 *)

for(n=1,10^7, if(3*numdiv(n)==eulerphi(n),print(n)))

"full" keyword from Max Alekseyev, Mar 01 2010

A063470 Numbers n such that tau(n)*4 = phi(n).

Original entry on oeis.org

34, 45, 52, 102, 140, 156, 252, 360, 420

Offset: 1

Jason Earls, Jul 26 2001

Phi(n) = k*tau(n) has at most finitely many solutions for any constant k or even for slowly increasing functions like k(n) = n^(epsilon). - Labos Elemer, Jul 20 2001

For n > 2, tau(n) > 2 and phi(n) <= n-1 so the least solution a(1) to tau(n)*k = phi(n), must be a(1) >= 2*k+1, for the case k=4, a(1) >= 2*4+1 = 9. - Enrique Pérez Herrero, May 12 2012

Cf. A062516.

Cf. A112954, A020488, A063469.

for(n=1,10^6, if(numdiv(n)*4==eulerphi(n),print(n)))

a(1) = A175667(4)

a(A112954(4)) = A112955(4). - Enrique Pérez Herrero, May 12 2012

"full" keyword from Max Alekseyev, Mar 01 2010

A063070 a(n) = phi(n) - d(n), where d(n) is the number of divisors function (A000005).

Original entry on oeis.org

0, -1, 0, -1, 2, -2, 4, 0, 3, 0, 8, -2, 10, 2, 4, 3, 14, 0, 16, 2, 8, 6, 20, 0, 17, 8, 14, 6, 26, 0, 28, 10, 16, 12, 20, 3, 34, 14, 20, 8, 38, 4, 40, 14, 18, 18, 44, 6, 39, 14, 28, 18, 50, 10, 36, 16, 32, 24, 56, 4, 58, 26, 30, 25, 44, 12, 64, 26, 40, 16, 68, 12, 70, 32, 34, 30, 56, 16, 76, 22, 49, 36, 80, 12, 60, 38, 52, 32, 86, 12, 68, 38

Offset: 1

Jason Earls, Aug 04 2001

It is known that a(n) >= 1 for n >= 31.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000005, A000010 (phi). A020488 gives n such that a(n) = 0.

Table[EulerPhi[n] - DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Nov 24 2021 *)

j=[]; for(n=1,150,j=concat(j,eulerphi(n)-(numdiv(n)))); j

{ for (n=1, 1000, write("b063070.txt", n, " ", eulerphi(n) - numdiv(n)) ) } \\ Harry J. Smith, Aug 16 2009

a(n) = A000010(n) - A000005(n). - Wesley Ivan Hurt, Nov 24 2021

a(n) = Sum_{d|n} (d*mu(n/d) - 1). - Wesley Ivan Hurt, Jul 21 2025

A083243 Numbers k for which there are more divisors and coprimes than other numbers less than k: A045763(k) < A073757(k) or A045763(k) < k/2 or A073757(k) > k/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76

Offset: 1

Labos Elemer, May 07 2003

nonn

			30 is not in the sequence because d(30) + phi(30) - 1 = 8 + 8 - 1 = 15. There are as many divisors and coprimes as there are numbers j <= 30 that neither divide nor are coprime to 30.
50 is not here because d(50) + phi(50) - 1 = 6 + 20 - 1 = 25. There are as many divisors and coprimes as there are numbers j < 50 that neither divide nor are coprime to 50.
146 is here because d(146) + phi(146) - 1 = 4 + 72 - 1 = 75; 146/2 = 73, and 75 > 73.
61455 is here because d(61455) + phi(61455) - 1 = 16 + 30720 - 1 = 30735; 61455/2 = 30727 + 1/2, and 30735 > 61455/2.

Cf. A000005, A000010, A045763, A073757, A020488, A083244, A083245.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[n-u, n/2], Print[n, {d, r, u}]], {n, 1, 100}]
(* Second program: *)
Select[Range[120], DivisorSigma[0, #] + EulerPhi[#] - 1 > #/2 &] (* Michael De Vlieger, Aug 22 2023 *)

{ k : d(k) + phi(k) - 1 > k/2 }.

Data corrected and entry edited by Michael De Vlieger, Aug 22 2023

A112955 Greatest number m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005; a(n)=0 if no such m exists.

Original entry on oeis.org

30, 120, 210, 420, 330, 840, 294, 1260, 1080, 1320, 690, 2520, 318, 1470, 2310, 3360, 0, 3780, 0, 4620, 1290, 2760, 1410, 5460, 3000, 1590, 7560, 5880, 1770, 9240, 0, 10080, 4830, 1236, 3234, 10920, 894, 0, 2370, 13860, 2490, 6090, 1038, 9660, 11880

Offset: 1

Reinhard Zumkeller, Oct 07 2005

nonn

Each term is a multiple of 6. [Max Alekseyev, Mar 01 2010]

			a(1) = A020488(A112954(1)) = A020488(7) = 30;
a(2) = A062516(A112954(2)) = A062516(9) = 120;
a(3) = A063469(A112954(3)) = A063469(10) = 210;
a(4) = A063470(A112954(4)) = A063470(9) = 420.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A112954.

Cf. A175667. [Enrique Pérez Herrero, Oct 22 2010]

More terms from Max Alekseyev, Mar 01 2010

A175667 Smallest number m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005; a(n)=0 if no such m exists.

Original entry on oeis.org

1, 5, 7, 34, 11, 13, 58, 17, 19, 55, 23, 65, 106, 29, 31, 85, 0, 37, 0, 41, 43, 115, 47, 119, 125, 53, 133, 145, 59, 61, 0, 388, 67, 274, 71, 73, 298, 0, 79, 187, 83, 203, 346, 89, 209, 235, 0, 97, 394, 101, 103, 169, 107, 109, 253, 113, 458, 295, 0, 287, 0, 0, 127, 514, 131

Offset: 1

Enrique Pérez Herrero, Aug 05 2010

nonn

If p = 2*n+1 is a prime, and if n > 1 then a(n)=p.
From R. J. Mathar, Aug 07 2010: (Start)
First column in the array
1,3,8,10,18,24,30: A020488
5,9,15,28,40,72,84,90,120: A062516
7,21,26,56,70,78,108,126,168,210: A063469
34,45,52,102,140,156,252,360,420: A063470
11,33,88,110,198,264,330,
13,35,39,63,76,104,105,130,228,234,280,312,390,504,540,630,840,
58,98,174,294,
17,51,128,136,170,176,224,260,306,384,408,468,510,528,672,780,1260,
19,57,74,135,152,182,190,222,342,456,546,570,756,1080,
55,82,99,124,165,246,308,350,372,440,792,924,990,1050,1320,
23,69,184,230,414,552,690,
65,117,148,195,238,315,364,380,444,520,684,714,864,936,1092,1140,1170,1560,2520,
... (End)

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A112954, A112955

Cf. A005384, A023212, A043297.

Table[SelectFirst[Range[10^5], EulerPhi@ # == n DivisorSigma[0, #] &] /.
k_ /; MissingQ@ k -> 0, {n, 120}] (* Michael De Vlieger, Aug 09 2017, Version 10.2 *)

From Enrique Pérez Herrero, Jan 01 2012: (Start)
If n > 1 then a(n) >= 2*n+1 or a(n)=0.
If p and q = 2*p+1 are both prime, A005384, then a(p) = 2*p+1.
If p > 3 and q = 4*p+1 are both prime, A023212, then a(p) = 8*p + 2 = 2*q.
If p > 2 is prime and both 2*p+1 and 4*p+1 are composite, A043297, then a(n)=0.
(End)

More terms from R. J. Mathar, Aug 07 2010

Comment corrected by Enrique Pérez Herrero, Aug 12 2010

A083244 k is in the sequence iff the number of numbers unrelated to k is larger than that of related ones[=divisors and coprimes] to k: A045763(k) > A073757(k) or A045763(k) > k/2 or A073757(k) < k/2.

Original entry on oeis.org

42, 54, 60, 66, 70, 72, 78, 84, 90, 96, 98, 100, 102, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 170, 174, 180, 182, 186, 190, 192, 196, 198, 200, 204, 210, 216, 220, 222, 224, 228, 230, 234, 238, 240, 242, 246, 250, 252

Offset: 1

Labos Elemer, May 07 2003

nonn

			k = 42 is a term because d = 8 divisors, r = 12 coprimes and u = 23 unrelated belong to it: u = 23 > 19 = 8 + 12 - 1 = d + r - 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A045763, A073757, A020488, A083243, A083245.

filter:= n -> n > 2*(numtheory:-tau(n) + numtheory:-phi(n)-1):
select(filter, [$1..1000]); # Robert Israel, May 15 2017

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-d-r+1; If[Greater[u, n/2], Print[n, {d, r, u}]], {n, 1, 100}]
(* Second program: *)
Select[Range@ 256, # - (DivisorSigma[0, #] + EulerPhi[#] - 1) > #/2 &] (* Michael De Vlieger, Jul 22 2017 *)

Numbers k such that k - d(k) - phi(k) + 1 > k/2.

cp's OEIS Frontend

A064435 Duplicate of A020488.

Views

Author

Keywords

A062516 Numbers k such that 2*tau(k) = phi(k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A112954 Number of numbers m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005.

Views

Author

Keywords

Links

Crossrefs

Extensions

A063469 Numbers n such that tau(n)*3 = phi(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A063470 Numbers n such that tau(n)*4 = phi(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

Extensions

A063070 a(n) = phi(n) - d(n), where d(n) is the number of divisors function (A000005).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A083243 Numbers k for which there are more divisors and coprimes than other numbers less than k: A045763(k) < A073757(k) or A045763(k) < k/2 or A073757(k) > k/2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A112955 Greatest number m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005; a(n)=0 if no such m exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A175667 Smallest number m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005; a(n)=0 if no such m exists.

Views

Author

Keywords

Comments

Links