A020491 -id:A020491 - cp's OEIS frontend

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

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

A015733 Numbers k such that d(k) does not divide phi(k).

Original entry on oeis.org

2, 4, 6, 12, 14, 16, 20, 22, 25, 27, 32, 36, 38, 42, 44, 46, 48, 50, 54, 60, 62, 64, 66, 68, 75, 80, 81, 86, 92, 94, 96, 100, 112, 114, 116, 118, 121, 132, 134, 138, 142, 144, 150, 154, 158, 160, 162, 164, 166, 180, 186, 188, 189, 192, 196, 200, 204

Offset: 1

Robert G. Wilson v

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..18157 (first 2000 terms from Enrique Pérez Herrero)

Cf. A000005 (d), A000010 (phi), A015734, A020491 (d(k) does divide phi(k)).

Select[Range[204], ! Divisible[EulerPhi[#], DivisorSigma[0, #]] &]

is(k) = {my(f = factor(k), d = numdiv(f), p = eulerphi(f)); p % d;} \\ Amiram Eldar, May 15 2024

A279289 Numbers k such that phi(k) > tau(k).

Original entry on oeis.org

5, 7, 9, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Jaroslav Krizek, Dec 09 2016

nonn

Numbers k such that A000010(k) > A000005(k).
There are 11 numbers k such that phi(k) <= tau(k) and 7 numbers k such that phi(k) = tau(k); see A020490 and A020488.
For k >= 31; phi(k) - tau(k) >= 1, see A063070.

			14 is a term because phi(14) = 6 > tau(14) = 4.

Complement of A020490.

Cf. A000005, A000010, A020488, A020491, A063070, A279287, A279288.

[n: n in[1..1000] | EulerPhi(n) gt NumberOfDivisors(n)];

Select[Range@ 77, EulerPhi@ # > DivisorSigma[0, #] &] (* Michael De Vlieger, Dec 11 2016 *)

is(n) = eulerphi(n) > numdiv(n) \\ Felix Fröhlich, Dec 09 2016

a(n)=if(n<20, select(k -> eulerphi(k)>numdiv(k), [5..29])[n], n+11) \\ Charles R Greathouse IV, Dec 16 2016

a(n) = n + 11 for n >= 20.

A289585 Quotients as they appear as k increases when tau(k) divides phi(k).

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 1, 5, 6, 2, 8, 1, 9, 3, 11, 1, 3, 2, 14, 1, 15, 5, 4, 6, 18, 6, 2, 20, 21, 4, 23, 14, 8, 4, 26, 10, 3, 9, 7, 29, 30, 6, 12, 33, 11, 3, 35, 2, 36, 9, 6, 15, 3, 39, 10, 41, 2, 16, 14, 5, 44, 2, 18, 15, 18, 48, 7, 10, 50, 4, 51, 6, 6, 13, 53, 3, 54, 5, 18, 56, 22, 12, 24, 2

Offset: 1

Bernard Schott, Jul 08 2017

nonn

Numbers k such that tau(k) divides phi(k) are in A020491.
Only for seven integers which are in A020488, we have a(n) = 1.
The integers such that a(n) = 2, 3, 4 are respectively in A062516, A063469, A063470.
When p is an odd prime then phi(p) = p-1, tau(p) = 2, so phi(p)/tau(p) = (p-1)/2 and A005097 is an infinite subsequence.
For k = A058891(m+1), that is 2^A000225(m), with m>=2, the corresponding quotient phi(k)/tau(k) is the integer A076688(m). - Bernard Schott, Aug 15 2020

			a(10) = 2 because A020491(10) = 15 and phi(15)/tau(15) = 8/4 = 2.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A000010, A000005, A020491, A015733, A020488, A058891, A062516, A063469, A063470, A175667, A112955, A112954.
Cf. A000225, A058891, A076688.
Cf. A005097 (a subsequence).
Cf. A290634 (complement).

for n from 1 to 50 do q:=phi(n)/tau(n);
if q=floor(q) then print(n,q,phi(n),tau(n)) else fi; od:

f[n_] := Block[{d = EulerPhi[n]/DivisorSigma[0, n]}, If[ IntegerQ@d, d, Nothing]]; Array[f, 120] (* Robert G. Wilson v, Jul 09 2017 *)

lista(nn) = {for (n=1, nn, q = eulerphi(n)/numdiv(n); if (denominator(q)==1, print1(q, ", ")););} \\ Michel Marcus, Jul 10 2017

a(n) = A000010(A020491(n)) / A000005(A020491(n)). - David A. Corneth, Jul 09 2017

A124331 a(n) is the ((phi(n) mod d(n)) + 1)-th positive divisor of n, where phi(n) is number of positive integers which are <= n and are coprime to n and d(n) is the number of positive divisors of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 3, 1, 1, 1, 1, 1, 6, 1, 7, 1, 8, 1, 1, 1, 4, 1, 11, 1, 1, 25, 1, 9, 1, 1, 1, 1, 16, 1, 1, 1, 4, 1, 19, 1, 1, 1, 7, 1, 4, 1, 23, 1, 12, 1, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 31, 1, 16, 1, 11, 1, 4, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 4, 81, 1, 1, 1, 1, 43, 1, 1, 1, 1, 1, 4, 1, 47, 1, 24, 1, 1, 1

Offset: 1

Leroy Quet and Ray Chandler, Oct 26 2006

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A000010, A124219, A124330.

Cf. A020491 (positions of 1's), A342665 (fixed points).

f[n_] := Block[{d = Divisors[n]}, d[[Mod[EulerPhi[n], Length[d]] + 1]]];Table[f[n], {n, 100}] (* Ray Chandler, Oct 26 2006 *)

A124331(n) = { my(m=eulerphi(n)%numdiv(n), ds=divisors(n)); ds[1+m]; }; \\ Antti Karttunen, Mar 30 2021

A279287 a(n) = numerator of (phi(n)/tau(n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 1, 2, 1, 5, 2, 6, 3, 2, 8, 8, 1, 9, 4, 3, 5, 11, 1, 20, 3, 9, 2, 14, 1, 15, 8, 5, 4, 6, 4, 18, 9, 6, 2, 20, 3, 21, 10, 4, 11, 23, 8, 14, 10, 8, 4, 26, 9, 10, 3, 9, 7, 29, 4, 30, 15, 6, 32, 12, 5, 33, 16, 11, 3, 35, 2, 36, 9, 20, 6, 15, 3

Offset: 1

Jaroslav Krizek, Dec 09 2016

a(n) = numerator of (A000010(n)/A000005(n)).
See A279288 (denominator of (phi(n)/tau(n))) and A063070 (phi(n)-tau(n)).
a(n) = 1 and A279288(n) = 1 for numbers n in A020488; a(n) > A279288(n) for numbers n in A279289.

			For n = 6: phi(6)/tau(6) = 2/4 = 1/2; a(6) = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A020488, A020490, A020491, A063070, A279288, A279289.

[Numerator(EulerPhi(n)/NumberOfDivisors(n)): n in[1..1000]];

with(numtheory): A279287:=n->numer(phi(n)/sigma(n)): seq(A279287(n), n=1..150); # Wesley Ivan Hurt, Dec 11 2016

Table[Numerator[EulerPhi[n]/DivisorSigma[0, n]], {n, 78}] (* Michael De Vlieger, Dec 09 2016 *)

a(n) = numerator(eulerphi(n)/numdiv(n)) \\ Felix Fröhlich, Dec 09 2016

A279288 a(n) = denominator of (phi(n)/tau(n)).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 5, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 5, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 2, 1

Offset: 1

Jaroslav Krizek, Dec 09 2016

a(n) = denominator of (A000010(n)/A000005(n)).
See A279287 (numerator of (phi(n)/tau(n))) and A063070 (phi(n)-tau(n)).
a(n) = 1 and A279287(n) = 1 for numbers n in A020488; A279287(n) > a(n) for numbers n in A279289.

			For n = 6: phi(6)/tau(6) = 2/4 = 1/2; a(6) = 2.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A020488, A020490, A020491, A063070, A279287, A279289.

[Denominator(EulerPhi(n)/NumberOfDivisors(n)): n in[1..1000]];

Table[Denominator[EulerPhi[n]/DivisorSigma[0, n]], {n, 120}] (* Michael De Vlieger, Dec 10 2016 *)

a(n) = denominator(eulerphi(n)/numdiv(n)) \\ Felix Fröhlich, Dec 09 2016

a(n) = 1 for numbers in A020491.

A286627 a(n) = exponent of the highest power of A000005(n) (number of divisors of n) dividing A000010(n) (totient function phi), a(1) = 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 4, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 2, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 1, 3, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 1, 3, 0, 1, 1, 3, 1, 1, 0, 1, 0, 1, 0, 5, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 4, 0, 1, 0, 2, 0, 2, 1, 0

Offset: 1

Antti Karttunen, Jun 30 2017

nonn

a(1) = 1 by convention.

			A000005(5) = 2, A000010(5) = 4, 2^2 is the highest power of 2 which divides 4, thus a(5) = 2.
A000005(6) = 4, A000010(6) = 2, 4^0 = 1 is the highest power of 4 which divides 2, thus a(6) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A286561, A286628, A289276.

Cf. A015733 (positions of zeros), A020491 (of nonzeros).

A286627(n) = valuation(eulerphi(n), numdiv(n));

a(n) = A286561(A000010(n), A000005(n)).

A020493 Numbers k such that d(k) (number of divisors) divides phi(k) (Euler function) divides sigma(k) (sum of divisors).

Original entry on oeis.org

1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 357, 420, 570, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1485, 1672, 1848, 2090, 2214, 2436, 2580, 2730, 3080, 3135, 3339, 3596, 3720, 3956, 4064, 4180, 4522, 4674, 5016, 5049, 5278, 5396

Offset: 1

David W. Wilson

nonn

Numbers k such that sigma_0(k) divides phi(k) divides sigma_1(k).

			210 has 16 divisors, which divides phi(210) = 48, which in turn divides sigma(210) = 576, so 210 is a term of the sequence.

David Wells, Curious and interesting numbers, Penguin Books, p. 130.

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

Cf. A000005, A000010, A000203.

Intersection of A020491 and A020492.

q[n_] := And @@ IntegerQ /@ Ratios @ {DivisorSigma[0, n], EulerPhi[n], DivisorSigma[1, n]}; Select[Range[6000], q] (* Amiram Eldar, Apr 13 2024 *)

for(n=1, 1e3, if(sigma(n)%eulerphi(n)==0, if(sigma(n)%numdiv(n)==0, if(eulerphi(n)%numdiv(n)==0, print1(n, ", "))))) \\ Felix Fröhlich, Aug 08 2014

Wells incorrectly has 52 instead of 56.

A069237 Composite numbers k such that tau(k) divides phi(k), where tau(k) is the number of divisors of k and phi(k) the Euler totient function.

Original entry on oeis.org

8, 9, 10, 15, 18, 21, 24, 26, 28, 30, 33, 34, 35, 39, 40, 45, 49, 51, 52, 55, 56, 57, 58, 63, 65, 69, 70, 72, 74, 76, 77, 78, 82, 84, 85, 87, 88, 90, 91, 93, 95, 98, 99, 102, 104, 105, 106, 108, 110, 111, 115, 117, 119, 120, 122, 123, 124, 125, 126, 128, 129, 130, 133

Offset: 1

Benoit Cloitre, Apr 13 2002

Includes A046388 and 2*A002144. - Robert Israel, Jan 05 2018

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

Composite numbers in A020491.

Cf. A002144, A046388.

filter:= n -> not isprime(n) and numtheory:-phi(n) mod numtheory:-tau(n)=0:
select(filter, [$4..1000]); # Robert Israel, Jan 05 2018

nn=200;Rest[Select[Complement[Range[nn],Prime[Range[PrimePi[nn]]]],Divisible[ EulerPhi[#], DivisorSigma[0,#]]&]] (* Harvey P. Dale, Mar 31 2011 *)

isok(k) = if(k == 1 || isprime(k), 0, my(f = factor(k)); !(eulerphi(f) % numdiv(f))); \\ Amiram Eldar, Apr 19 2025

cp's OEIS Frontend

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

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A015733 Numbers k such that d(k) does not divide phi(k).

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A279289 Numbers k such that phi(k) > tau(k).

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A289585 Quotients as they appear as k increases when tau(k) divides phi(k).

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A124331 a(n) is the ((phi(n) mod d(n)) + 1)-th positive divisor of n, where phi(n) is number of positive integers which are <= n and are coprime to n and d(n) is the number of positive divisors of n.

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A279287 a(n) = numerator of (phi(n)/tau(n)).

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A279288 a(n) = denominator of (phi(n)/tau(n)).

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A286627 a(n) = exponent of the highest power of A000005(n) (number of divisors of n) dividing A000010(n) (totient function phi), a(1) = 1.

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