A020488 -id:A020488 - cp's OEIS frontend

A114063 Numbers k such that phi(k) = tau(k)^4, where tau(k) = A000005(k).

1, 17, 514, 8738, 32301, 33003, 36351, 41504, 42292, 43852, 51860, 62226, 549117, 561051, 571311, 599067, 617967, 629811, 634005, 657495, 673184, 674505, 683168, 701024, 705568, 718964, 722684, 732628, 745484, 759772, 774368

Offset: 1

Giovanni Resta, Feb 13 2006

For all large enough k, we have tau(k) < k^(1/5) and phi(k) > k^(4/5). Hence, tau(k)^4 < k^(4/5) < phi(k), implying that this sequence is finite. - Max Alekseyev, Mar 10 2016

Sequence is composed of 94030 terms. - Max Alekseyev, Jun 01 2024

			phi(33003) = 20736. tau(33003) = 12, 20736 = 12^4.
a(2) = A107655(4) = 17.

Max Alekseyev, Table of n, a(n) for n = 1..94030 (first 660 terms from Enrique Pérez Herrero; terms 661..7000 from Giovanni Resta)

Subsequence of A078164.

Cf. A020488, A062516, A063469, A063470, A107655, A112954, A112955, A175667, A068560, A068559.

Select[Range[10^6], EulerPhi[#] == DivisorSigma[0, #]^4 &] (* Paolo Xausa, May 31 2024 *)

isok(n) = eulerphi(n) == numdiv(n)^4; \\ Michel Marcus, Jan 22 2014

A068559 Numbers m such that phi(m) = tau(m)^3.

Original entry on oeis.org

1, 85, 333, 436, 1542, 1875, 2907, 3285, 3488, 3796, 5196, 10280, 17532, 17776, 20080, 21250, 28305, 30368, 30555, 32708, 34748, 35308, 36860, 37060, 41544, 41568, 43068, 44004, 45162, 48468, 51930, 81324, 98304, 98688, 104856, 131070

Offset: 1

Benoit Cloitre, Mar 25 2002

For all large enough k, we have tau(k) < k^(1/4) and phi(k) > k^(3/4). Hence, tau(k)^3 < k^(3/4) < phi(k), implying that this sequence is finite. In fact, the sequence consists of 614 terms. - Max Alekseyev, May 30 2024

			a(2) = A107655(3) = 85.

Max Alekseyev, Table of n, a(n) for n = 1..614 (first 470 terms from Enrique Pérez Herrero)

Subsequence of A039771. - Enrique Pérez Herrero, Aug 29 2010

Cf. A020488, A068560, A107655, A114063.

Select[Range[132000],EulerPhi[#]==DivisorSigma[0,#]^3&]  (* Harvey P. Dale, Dec 28 2022 *)

isok(m) = eulerphi(m) == numdiv(m)^3; \\ Michel Marcus, Oct 18 2019

A083245 Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2A045763(n) = 2A073757(n)-n.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 6, 7, 4, 11, 6, 13, 4, 7, 8, 17, 4, 19, 6, 9, 4, 23, 6, 19, 4, 15, 6, 29, 0, 31, 10, 13, 4, 19, 4, 37, 4, 15, 6, 41, -4, 43, 6, 13, 4, 47, 2, 39, 0, 19, 6, 53, -4, 31, 6, 21, 4, 59, -6, 61, 4, 19, 12, 37, -12, 67, 6, 25, -8, 71, -2, 73, 4, 15, 6, 49, -16, 79, 2, 35, 4, 83, -14, 49, 4, 31, 6, 89, -20, 59, 6, 33, 4, 55, -10, 97, -4

Offset: 1

Labos Elemer, May 07 2003

sign

There are only 2 cases [n=30, n=50] below 10^7 such that a(n) = 0.

No other zeros found up to 10^9. - Michel Marcus, Jul 30 2017

			n=37, d=2,r=36,u=0, a(37)=2+36-1-0=37>0; primes are fixed points.
n=42, d=8,r=12,u=23,a(42)=8+12-1-23=-4<0, terms of A083244;
n=30, d=8,r=8,u=15, a(30)=0;
n=50, d=6,r=20,u=25,a(50)=0.

Michel Marcus, Table of n, a(n) for n = 1..10000

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

Table[2*(DivisorSigma[0, w]+EulerPhi[w]-1)-w, {w, 1, 1000}]

a(n) = 2*(numdiv(n)+eulerphi(n)-1) - n; \\ Michel Marcus, Jul 30 2017

a(n) = 2(A000005(n)+A000010(n)-1)-n.

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

A083246 Numbers n such that at least one of the following four conditions is satisfied: 1# d(n)=phi(n); 2# d(n)=u(n); 3# phi(n)=u(n), or 4# n=2u(n). Here d(n)=A000005(n) is the number of divisors of n, phi(n)=A000010(n) is Euler's totient and u(n)=A045763(n) is the size of the 'unrelated set'.

Original entry on oeis.org

1, 3, 8, 10, 15, 18, 24, 25, 30, 50, 61455

Offset: 1

Labos Elemer, May 07 2003

Is this sequence complete?

			1# d(n)=phi(n) holds for {1,3,8,10,18,24,30}, see A020488;
2# d(n)=u(n) holds for {15,25};
3# phi(n)=u(n) holds for {61455};
4# n=2u(n) holds for {30,50}. No more cases below 10^7.
{n,d,r,u} values for 11 initial terms are as follows:
{1, 1, 1, 0}, {3, 2, 2, 0}, {8, 4, 4, 1}, {10, 4, 4, 3}, {15, 4, 8, 4}, {18, 6, 6, 7}{24, 8, 8, 9}, {25, 3, 20, 3}, {30, 8, 8, 15}, {50, 6, 20, 25}, {61455, 16, 30720, 30720}.

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

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; If[Equal[d, r]||Equal[d, u]||Equal[r, u]||Equal[u, n-u], Print[n(*, {d, r, u}*)]], {n, 1, 10000000}]

is(n)=my(r=eulerphi(n),d=numdiv(n),u=n-r-d+1);d==r||d==u||r==u||2*u==n \\ Charles R Greathouse IV, Feb 21 2013

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.

A289276 Numbers k such that phi(k) (the totient function A000010) is a power of the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 17, 18, 24, 30, 34, 63, 76, 85, 128, 136, 170, 257, 315, 333, 364, 380, 436, 444, 514, 640, 680, 972, 1285, 1542, 1820, 1824, 1836, 1875, 2142, 2220, 2907, 3285, 3488, 3796, 4369, 4788, 4860

Offset: 1

Franklin T. Adams-Watters, Jun 30 2017

A019434 is a subsequence. - David A. Corneth, Jun 30 2017

Is the frequency of e such that A000005(a(n))^e = A000010(a(n)) finite? - David A. Corneth, Jul 01 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..5319 (first 526 terms from Antti Karttunen)

Cf. A000005, A000010, A019434, A020488, A032447, A036913, A051281, A068559, A068560, A114063, A286627.

Join[{1},Select[Range[2,5000],IntegerQ[Log[DivisorSigma[0,#],EulerPhi[#]]]&]] (* Harvey P. Dale, Aug 06 2017 *)

ispowerof(n, k)= if(k==1, return(n==1)); while(n>=k, if(n%k!=0, return(0)); n\=k); n==1
isa(n) = ispowerof(eulerphi(n),numdiv(n)) \\ Quick program, fast enough for early values.

is(n) = if(n==1, return(1)); my(f = factor(n); phi = eulerphi(f), ndiv = numdiv(f), e = logint(phi, ndiv)); ndiv^e == phi \\ David A. Corneth, Jun 30 2017, changed per suggestion of Charles R Greathouse IV

isA289276(n)= if(n==1, return(1)); my(phi = eulerphi(n), ndiv = numdiv(n), v = valuation(phi, ndiv)); ndiv^v == phi; \\ (A variant of above program). - Antti Karttunen, Jun 30 2017

list(lim)=my(v=List([1])); forfactored(n=2,lim\1, my(phi = eulerphi(n), ndiv = numdiv(n)); if(ndiv^valuation(phi,ndiv) == phi, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 01 2017

A112587 Numbers m such that phi(m) <= 2*tau(m), where phi=A000010 and tau=A000005.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 36, 40, 42, 48, 60, 72, 84, 90, 120

Offset: 1

Reinhard Zumkeller, Sep 14 2005

Complement of A112588.

Eric Weisstein's World of Mathematics, Totient Function
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A020488.

Select[Range[150],EulerPhi[#]<=2DivisorSigma[0,#]&] (* Harvey P. Dale, Mar 04 2013 *)

is(n)=eulerphi(n)<=2*numdiv(n) \\ Charles R Greathouse IV, Feb 21 2013

cp's OEIS Frontend

A114063 Numbers k such that phi(k) = tau(k)^4, where tau(k) = A000005(k).

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A068559 Numbers m such that phi(m) = tau(m)^3.

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A083245 Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2*A045763(n) = 2*A073757(n)-n.

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

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Magma

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

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A083246 Numbers n such that at least one of the following four conditions is satisfied: 1# d(n)=phi(n); 2# d(n)=u(n); 3# phi(n)=u(n), or 4# n=2u(n). Here d(n)=A000005(n) is the number of divisors of n, phi(n)=A000010(n) is Euler's totient and u(n)=A045763(n) is the size of the 'unrelated set'.

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

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A083245 Difference between numbers of related and numbers of unrelated numbers belonging to n: a(n) = A073757(n)-A045763(n) = (n-u(n))-u(n) = n-2A045763(n) = 2A073757(n)-n.