A020488 -id:A020488 - cp's OEIS frontend

A279507 a(n) = floor(phi(n)/tau(n)).

1, 0, 1, 0, 2, 0, 3, 1, 2, 1, 5, 0, 6, 1, 2, 1, 8, 1, 9, 1, 3, 2, 11, 1, 6, 3, 4, 2, 14, 1, 15, 2, 5, 4, 6, 1, 18, 4, 6, 2, 20, 1, 21, 3, 4, 5, 23, 1, 14, 3, 8, 4, 26, 2, 10, 3, 9, 7, 29, 1, 30, 7, 6, 4, 12, 2, 33, 5, 11, 3, 35, 2, 36, 9, 6, 6, 15, 3, 39, 3

Offset: 1

Jaroslav Krizek, Dec 13 2016

nonn

a(n) = floor(A000010(n)/A000005(n)).
There are 11 numbers n such that phi(n) <= tau(n) and 7 numbers n such that phi(n) = tau(n); see A020490 and A020488.
Sequences b(k) of numbers n such that a(n) = k are finite for all k >=0; see A279508 (the smallest numbers n such that a(n) = k for k>=0) and A279509 (the largest numbers n such that a(n) = k for k>=0).
See A140475 (numbers n such that floor(phi(n)/tau(n)) > floor(phi(m)/tau(m)) for all m < n).

			For n=5; a(5) = floor(phi(5)/tau(5)) = floor(4/2) = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..2000

Cf. A000005, A000010, A020488, A020490, A140475, A279289, A279508, A279509.

[Floor(EulerPhi(n)/NumberOfDivisors(n)): n in[1..100]]

Table[Floor[EulerPhi[n]/DivisorSigma[0, n]], {n,1,25}] (* G. C. Greubel, Dec 13 2016 *)

for(n=1, 25, print1(floor(eulerphi(n)/numdiv(n)), ", ")) \\ G. C. Greubel, Dec 13 2016

a(n) > 1 for numbers in A279289.

A083247 Numbers k such that A000010(k) > A045763(k) > A000005(k).

Original entry on oeis.org

14, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 49, 51, 52, 55, 57, 58, 62, 63, 64, 65, 68, 69, 74, 75, 76, 77, 81, 82, 85, 86, 87, 91, 92, 93, 94, 95, 99, 106, 111, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 128, 129, 133, 134, 135, 141, 142

Offset: 1

Labos Elemer, May 07 2003

Primes are not terms since A045763(p) = 0 < A000005(p) = 2 for a prime p.

			k = 99 is a term since d(k) = 6, phi(k) = 60, unrelateds(k) = 99 - 6 - 60 + 1 = 34, and 60 > 34 > 6 holds.

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

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083245, A083246, A083248, A020488.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[r, u]&&Greater[u, d], Print[n, {d, r, u}]], {n, 1, 1000}]

is(n)=my(r=eulerphi(n),d=numdiv(n),u=n-r-d+1); r>u && u>d \\ Charles R Greathouse IV, Feb 21 2013

A083248 Numbers k such that A045763(k) > A000010(k) > A000005(k).

Original entry on oeis.org

36, 40, 42, 48, 50, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186, 190, 192, 196, 198, 200, 204, 208, 210

Offset: 1

Labos Elemer, May 07 2003

nonn

Primes are not terms since A045763(p) = 0 < A000005(p) = 2 for a prime p.

			k = 100 is a term since d(k) = 9, phi(k) = 40, unrelateds(k) = 100 - 9 - 40 + 1 = 52, and 52 > 40 > 9 holds.

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

Cf. A000005, A000010, A045763, A073757, A083243, A083244, A083245, A083246, A083247, A020488.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Greater[u, r]&&Greater[r, d], Print[n, {d, r, u}]], {n, 1, 1000}]

isok(k) = {my(f = factor(k), d = numdiv(f), r = eulerphi(f), u = k - r - d + 1); u > r && r > d;} \\ Amiram Eldar, Feb 08 2025

A279508 a(n) = smallest number k such that floor(phi(k)/tau(k)) = n.

Original entry on oeis.org

2, 1, 5, 7, 27, 11, 13, 58, 17, 19, 55, 23, 65, 106, 29, 31, 85, 142, 37, 158, 41, 43, 115, 47, 119, 125, 53, 133, 145, 59, 61, 254, 262, 67, 274, 71, 73, 298, 1180, 79, 187, 83, 203, 346, 89, 209, 235, 382, 97, 394, 101, 103, 169, 107, 109, 253, 113, 458, 295

Offset: 0

Jaroslav Krizek, Dec 13 2016

nonn

a(n) = the smallest number k such that floor(A000010(k)/A000005(k)) = A279507(k) = n.
Sequences b_n of numbers k such that floor(phi(k)/tau(k)) = n for n = 0..2:
b_0: 2, 4, 6, 12;
b_1: 1, 3, 8, 10, 14, 16, 18, 20, 24, 30, 36, 42, 48, 60;
b_2: 5, 9, 15, 22, 28, 32, 40, 54, 66, 72, 84, 90, 96, 120, 180.
Sequences b_n are finite for all n >=0. See A279509 (largest number k such that floor(phi(k)/tau(k)) = n).
Supersequence of A045344 (primes excluding 3).

			For n = 2; a(2) = 5 because 5 is the smallest number with floor(phi(5) / tau(5)) = floor(4/2) = 2.

Cf. A000005, A000010, A020488, A020490, A045344, A279289, A279507, A279509.

[Min([n: n in[1..100000] | Floor(EulerPhi(n)/NumberOfDivisors(n)) eq k]): k in [0..60]]

Table[k = 1; While[Floor[EulerPhi[k]/DivisorSigma[0, k]] != n, k++]; k, {n, 0, 58}] (* Michael De Vlieger, Dec 14 2016 *)

a(n) = my(k=1); while(floor((eulerphi(k)/numdiv(k)))!=n, k++); k \\ Felix Fröhlich, Dec 14 2016

a((p-1)/2) = p for p = prime > 3.

A279509 a(n) = largest number k such that floor(phi(k)/tau(k)) = n.

Original entry on oeis.org

12, 60, 180, 240, 420, 480, 840, 462, 1260, 1680, 1440, 690, 2520, 2100, 2160, 2310, 3360, 2400, 3780, 5040, 4620, 3600, 3300, 1410, 5460, 4080, 6300, 7560, 5880, 4140, 9240, 2646, 10080, 6600, 6480, 7200, 10920, 8820, 9360, 2370, 13860, 8640, 8160, 15120

Offset: 0

Jaroslav Krizek, Dec 19 2016

nonn

a(n) = largest number k such that floor(A000010(k)/A000005(k)) = A279507(k) = n.
Sequences b_n of numbers k such that floor(phi(k)/tau(k)) = n for n = 0..2:
b_0: 2, 4, 6, 12;
b_1: 1, 3, 8, 10, 14, 16, 18, 20, 24, 30, 36, 42, 48, 60;
b_2: 5, 9, 15, 22, 28, 32, 40, 54, 66, 72, 84, 90, 96, 120, 180.
Sequences b_n are finite for all n >= 0. See A279508 (smallest number k such that floor(phi(k)/tau(k)) = n).

			For n = 1; a(1) = 60 because 60 is the largest number with floor(phi(60)/tau(60)) = floor(16/12) = 1.

Cf. A000005, A000010, A020488, A020490, A279289, A279507, A279508.

[Max([n: n in[1..100000] | Floor(EulerPhi(n) / NumberOfDivisors(n)) eq k]): k in [0..50]]

A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 3, 8, 10, 18, 19, 24, 30, 34, 45, 52, 54, 57, 73, 74, 85, 102, 125, 135, 140, 152, 153, 156, 163, 182, 185, 190, 202, 219, 222, 252, 255, 333, 342, 360, 375, 394, 416, 420, 436, 451, 455, 456, 459, 476, 489, 505, 514, 546, 555, 570, 584, 606, 625, 629, 640, 646, 679, 680, 730

Offset: 1

Bernard Schott, Feb 24 2021

nonn

The first 11 terms of this sequence are also the first 11 terms of A341939: m such that phi(m)/tau(m) is the square of an integer. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is a(12) = 54.
If k and q are terms and coprimes, then k*q is another term.
Some subsequences (see examples):
-> The seven terms that satisfy tau(m) = phi(m) form the subsequence A020488.
-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so tau(p)*phi(p) = (2*k)^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset with tau(p^3)*phi(p^3) = (2*k*p)^2.

			phi(18) = tau(18) = 6, so phi(18)*tau(18) = 6^2.
phi(19) = 18, tau(19) = 2, so phi(19)*tau(19) = 36 = 6^2.
phi(34) = 16, tau(34) = 4, so phi(34)*tau(34) = 16*4 = 64 = 8^2.
phi(125) = 100, tau(125) = 4, so phi(125)*tau(125) = 400 = 20^2.

Similar for: A011257 (phi*sigma square), A327830 (sigma*tau square).
Subsequences: A020488, A090698.
Cf. A341939, A341940.
Cf. A000005 (tau), A000010 (phi).

with(numtheory): filter:= n -> issqr(phi(n)*tau(n)) : select(filter, [$1..750]);

Select[Range[1000], IntegerQ @ GeometricMean[{DivisorSigma[0, #], EulerPhi[#]}] &] (* Amiram Eldar, Feb 24 2021 *)

isok(m) = issquare(numdiv(m)*eulerphi(m)); \\ Michel Marcus, Feb 24 2021

A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 3, 8, 10, 18, 19, 24, 30, 34, 45, 52, 57, 73, 74, 85, 102, 125, 135, 140, 152, 153, 156, 163, 182, 185, 190, 202, 219, 222, 252, 255, 333, 342, 360, 375, 394, 416, 420, 436, 451, 455, 456, 459, 476, 489, 505, 514, 546, 555, 570, 584, 606, 625, 629, 640, 646, 679, 680, 730

Offset: 1

Bernard Schott, Feb 24 2021

nonn

The first 11 terms of this sequence are also the first 11 terms of A341938: m such that phi(m)*tau(m) is a square, then, a(12) = 57 while A341938(12) = 54. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is 54 (last example).
Some subsequences (see examples):
-> The seven terms that satisfy also tau(m) = phi(m) form the subsequence A020488 with phi(m)/tau(m) = 1^2.
-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so phi(p)/tau(p) = k^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset because if p = 2, phi(8)/tau(8)=1, and if p odd, phi(p^3)/tau(p^3) = (k*p/2)^2 with k even.

			phi(30) = 8, tau(30) = 8 so phi(30)/tau(30) = 1^2, and 30 is a term.
phi(45) = 24, tau(45) = 6, so phi(45)/tau(45) = 4 = 2^2, and 85 is a term.
phi(125) = 100, tau(125) = 4, so phi(125)/tau(125) = 25 = 5^2, and 125 is a term.
phi(54) = 18, tau(54) = 8, and phi(54)/tau(54) = 18/8 = 9/4 = (3/2)^2 and 54 is not a term while phi(54)*tau(54) = 12^2.

Intersection of A020491 and A341938.
Similar for: A144695 (sigma(n)/tau(n) perfect square), A293391 (sigma(n)/phi(n) perfect square).
Subsequences: A090698, A020488.
Cf. A069237, A289585, A341940.
Cf. A000005 (phi), A000010(tau).

with(numtheory): filter:= q -> phi(q)/tau(q) = floor(phi(q)/tau(q)) and issqr(phi(q)/tau(q)) : select(filter, [$1..750]);

Select[Range[1000], IntegerQ @ Sqrt[EulerPhi[#]/DivisorSigma[0, #]] &] (* Amiram Eldar, Feb 24 2021 *)

isok(m) = my(x=eulerphi(m)/numdiv(m)); (denominator(x)==1) && issquare(x); \\ Michel Marcus, Feb 24 2021

A055517 Largest integer k such that k | Phi(k) - d(k) + n, or 0 if no such k exists, where d() is the number of divisors.

Original entry on oeis.org

30, 4, 12, 0, 1, 4, 9, 2, 25, 4, 49, 15, 14, 27, 121, 35, 169, 33, 26, 55, 289, 77, 361, 91, 38, 85, 529, 143, 46, 133, 44, 187, 841, 221, 961, 247, 62, 253, 42, 323, 1369, 217, 74, 391, 1681, 437, 1849, 403, 86, 493, 2209

Offset: 0

Robert G. Wilson v, Jul 03 2000

nonn

In the case of n = 3, the condition is satisfied for all the primes.

			for a(2), 12 | [ Phi(12) + d(12) + 2 ].

Cf. A020488.

A290634 Positive integers which are never the quotient of phi(n)/tau(n).

Original entry on oeis.org

17, 19, 31, 38, 47, 59, 61, 62, 71, 85, 91, 101, 103, 107, 109, 118, 121, 133, 137, 149, 151, 157, 167, 181, 187, 197, 211, 217, 218, 223, 227, 229, 241, 247, 257, 259, 263, 266, 269, 271, 283, 289, 305, 311, 313, 314, 317, 327, 331, 334, 337, 347, 349, 353, 355, 361, 367

Offset: 1

Bernard Schott, Aug 08 2017

nonn

For phi(n)/tau(n) see A279287/A279288.
Numbers that do not appear in A175667.
The first nine terms of this sequence are exactly A119480(3) through A119480(11), and many other terms are common to these two sequences.

Cf. A000010, A000005, A020491, A015733, A020488, A062516, A175667, A112954, A112955, A119480, A289585, A279287/A279288.

A351561 a(n) = n + d(n) - phi(n), where d is the number of divisors function, and phi is the Euler totient function.

Original entry on oeis.org

1, 3, 3, 5, 3, 8, 3, 8, 6, 10, 3, 14, 3, 12, 11, 13, 3, 18, 3, 18, 13, 16, 3, 24, 8, 18, 13, 22, 3, 30, 3, 22, 17, 22, 15, 33, 3, 24, 19, 32, 3, 38, 3, 30, 27, 28, 3, 42, 10, 36, 23, 34, 3, 44, 19, 40, 25, 34, 3, 56, 3, 36, 33, 39, 21, 54, 3, 42, 29, 54, 3, 60, 3, 42, 41, 46, 21, 62, 3, 58, 32, 46, 3, 72, 25, 48

Offset: 1

Antti Karttunen, Feb 21 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000010 (phi), A008683 (mu), A020488 (fixed points), A051953, A063070.

Cf. also A055517.

Array[# + DivisorSigma[0, #] - EulerPhi[#] &, 86] (* Michael De Vlieger, Feb 21 2022 *)

PARI
```
A351561(n) = (n+numdiv(n)-eulerphi(n));
```

a(n) = A000005(n) + A051953(n) = n - A063070(n) = n + A000005(n) - A000010(n).

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

cp's OEIS Frontend

A279507 a(n) = floor(phi(n)/tau(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A083247 Numbers k such that A000010(k) > A045763(k) > A000005(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A083248 Numbers k such that A045763(k) > A000010(k) > A000005(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A279508 a(n) = smallest number k such that floor(phi(k)/tau(k)) = n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A279509 a(n) = largest number k such that floor(phi(k)/tau(k)) = n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

A055517 Largest integer k such that k | Phi(k) - d(k) + n, or 0 if no such k exists, where d() is the number of divisors.