A062821 -id:A062821 - cp's OEIS frontend

A163374 a(n) = tau(tau(phi(n))).

1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 3, 3, 3, 2, 3, 4, 3, 4, 3, 3, 3, 4, 4, 4, 4, 4, 3, 4, 2, 4, 2, 4, 4, 3, 4, 4, 2, 4, 4, 4, 4, 4, 3, 3, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 3, 2, 6, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 3, 4, 3, 6, 4, 4, 4, 4, 4, 3, 4, 2, 4

Offset: 1

Jaroslav Krizek, Jul 25 2009

			tau(tau(phi(7))) = tau(tau(6)) = tau(4) = 3. Thus a(7) = 3. - _Derek Orr_, Jul 27 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005 (tau), A000010 (phi), A010553, A062821.

[NumberOfDivisors(NumberOfDivisors(EulerPhi(n))): n in [1..100]]; // Vincenzo Librandi, Jul 27 2014

DivisorSigma[0,DivisorSigma[0,EulerPhi[Range[90]]]] (* Harvey P. Dale, Mar 25 2016 *)

a(n)=sigma(sigma(eulerphi(n),0),0); \\ Derek Orr, Jul 27 2014

a(n) = A000005(A000005(A000010(n))) = A000005(A062821(n)) = A010553(A000010(n)).

More terms from Vincenzo Librandi, Jul 27 2014

A163376 a(n) = phi(tau(phi(n))).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 4, 4, 2, 6, 2, 4, 4, 4, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 6, 2, 2, 4, 4, 4, 6, 2, 4, 2, 4, 2, 2, 4, 4, 4, 4, 6, 4, 6, 4, 4, 4, 2, 4, 4, 2, 4, 6, 4, 4, 4, 4, 4

Offset: 1

Jaroslav Krizek, Jul 25 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A062821, A163109.

with(numtheory): A163376:=n->phi(tau(phi(n))): seq(A163376(n), n=1..200); # Wesley Ivan Hurt, May 03 2017

EulerPhi[DivisorSigma[0, EulerPhi[Range[100]]]] (* G. C. Greubel, Dec 20 2016 *)

vector(100, n, eulerphi(numdiv(eulerphi(n)))) \\ G. C. Greubel, Dec 20 2016

a(n) = A000010(A000005(A000010(n))) = A000010(A062821(n)) = A163109(A000010(n)).

A173334 Numbers k such that tau(phi(k)) = phi(sum-of-prime-divisors(k)).

Original entry on oeis.org

2, 3, 15, 18, 24, 28, 30, 33, 39, 50, 52, 55, 80, 132, 133, 152, 169, 186, 187, 190, 195, 207, 215, 217, 222, 230, 238, 247, 261, 266, 305, 319, 333, 340, 352, 369, 371, 414, 481, 484, 494, 496, 497, 506, 516, 522, 559, 574, 580, 611, 644, 646, 660, 671, 689

Offset: 1

Michel Lagneau, Feb 16 2010

nonn

Numbers k such that A000005(A000010(k)) = A000010(A008472(k)).

			For n=15, tau(phi(15)) = tau(8)=4 equals phi(A008472(15))=phi(8) = 4, which adds 15 to the sequence.
For n=18, tau(phi(18)) = tau(6) =4 equals phi(A008472(18)) = phi(5) = 4, which adds 18 to the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.
W. Sierpinski, Number Of Divisors And Their Sum, Monogr. Matemat. 42 (1964) chapter IV
Wikipedia, Euler's totient function

Cf. A000005, A000010, A008472, A062821.

[m:m in [2..700]|#Divisors(EulerPhi(m)) eq EulerPhi(&+PrimeDivisors(m))]; // Marius A. Burtea, Jul 10 2019

with(numtheory): for n from 1 to 1800 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = phi(t2) then print (n): else fi : od :

Select[Range[2, 700], DivisorSigma[0, EulerPhi[#]] == EulerPhi[Total[FactorInteger[#][[All, 1]]]] &]
(* Jean-François Alcover, May 19 2011 *)

isok(n) = numdiv(eulerphi(n)) == eulerphi(vecsum(factor(n)[, 1])); \\ Michel Marcus, Jul 10 2019

{n : A062821(n)= phi(A008472(n))}.

Removed sopf acronym. Updated references and links - R. J. Mathar, Mar 10 2010

A173336 Numbers k such that tau(phi(k)) = sigma(sopf(k)).

Original entry on oeis.org

8, 9, 25, 36, 49, 54, 96, 100, 320, 441, 495, 704, 891, 1029, 1080, 1089, 1260, 1331, 1386, 1400, 1617, 1701, 1750, 1815, 1848, 1950, 1960, 2079, 2541, 2574, 2704, 2850, 2880, 3000, 3360, 3430, 3510, 3861, 4125, 4275, 4680, 4704, 4719, 4800, 5070, 5096

Offset: 1

Michel Lagneau, Feb 16 2010

nonn

tau(k) is the number of divisors of k (A000005); phi(k) is the Euler totient function (A000010); sigma(k) is the sum of divisors of k (A000203); and sopf(k) is the sum of the distinct primes dividing k without repetition (A008472).

			8 is in the sequence because phi(8) = 4, tau(4)=3, sopf(8)=2 and sigma(2) = 3 ;
9 is in the sequence because phi(9) = 6, tau(6)=4, sopf(9)=3 and sigma(3) = 4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.
Wikipedia, Euler's totient function

Cf. A000005, A000010, A000203, A001157, A001158, A001160, A001065, A002192, A008472, A062821.

[m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+Divisors(&+PrimeDivisors(m))]; // Marius A. Burtea, Jul 10 2019

with(numtheory): for n from 1 to 18000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = sigma(t2) then print (n): else fi : od :

sopf[n_] := Plus @@ (First@# & /@ FactorInteger[n]); Select[Range[2, 5100], DivisorSigma[0,EulerPhi[#]] == DivisorSigma[1, sopf[#]] &] (* Amiram Eldar, Jul 09 2019 *)

isok(n) = (n>1) && numdiv(eulerphi(n)) == sigma(vecsum(factor(n)[, 1])); \\ Michel Marcus, Jul 10 2019

k such that A062821(k) = sigma(A008472(k)).

Corrected and edited by Michel Lagneau, Apr 25 2010

Edited by D. S. McNeil, Nov 20 2010

A173618 Numbers k such that tau(phi(k)) = rad(k).

Original entry on oeis.org

1, 4, 36, 54, 96, 200, 448, 1280, 2700, 4500, 5103, 9720, 11264, 14112, 14580, 17280, 26624, 32928, 48000, 54432, 71442, 75000, 81648, 152064, 184320, 187500, 258048, 307200, 350000, 637875, 1250235, 1344560, 1557504, 2044416, 2187500, 2367488, 3234816

Offset: 1

Michel Lagneau, Feb 22 2010

nonn

rad(k) is the product of the primes dividing k (A007947), tau(k) is the number of divisors of k (A000005), phi(k) is the Euler totient function (A000010).

			phi(4) = 2, tau(2) = 2 and rad(4) = 2 phi(36) = 12, tau(12) = 6 and rad(36) = 6

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

Amiram Eldar, Table of n, a(n) for n = 1..100
W. Sierpinski, Number Of Divisors And Their Sum
Wikipedia, Euler's totient function

Cf. A000005, A000010, A062069, A062821, A007947, A173326.

with(numtheory):for n from 1 to 1000000 do : t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if tau(phi(n))= t2 then print (n): else fi: od :

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); Select[Range[10^5], DivisorSigma[0, EulerPhi[#]] == rad[#] &] (* Amiram Eldar, Jul 09 2019*)

isok(k) = numdiv(eulerphi(k)) == factorback(factorint(k)[, 1]); \\ Michel Marcus, Jul 09 2019

k such that A062821(k) = A007947(k).

a(30)-a(37) from Donovan Johnson, Jul 27 2011

A173745 Numbers n such that tau(phi(n)) = sigma(rad(n)).

Original entry on oeis.org

1, 8, 9, 25, 49, 216, 288, 324, 675, 1125, 1331, 1458, 1568, 2000, 2744, 3200, 3645, 6125, 6144, 8575, 9604, 9801, 14336, 30976, 31250, 42592, 46875, 70304, 72171, 81000, 108000, 109375, 123201, 129600, 131769, 135000, 145800, 182250, 184832

Offset: 1

Michel Lagneau, Feb 23 2010

nonn

tau(phi(n)) = A000005(A000010(n)) = A062821(n).

sigma(rad(n)) = A000203(A007947(n)) = A048250(n).

			For n=9, tau(phi(9)) = tau(6)=4 equals sigma(rad(9)) = sigma(3) = 4 which adds n=9 to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..100
W. Sierpinski, Number Of Divisors And Their Sum, Monogr. Matemat. 42 (1964) chapter IV

[1] cat [m:m in [2..200000]|#Divisors(EulerPhi(m)) eq &+Divisors(&*PrimeDivisors(m))]; // Marius A. Burtea, Jul 10 2019

with(numtheory):for n from 1 to 1500000 do : t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if tau(phi(n)) = sigma(t2) then print (n): else fi: od :

Select[Range[200000], DivisorSigma[0,EulerPhi[#]] == DivisorSigma[1, Times @@ FactorInteger[#][[All,1]]] & ] (* Jean-François Alcover, Sep 12 2011 *)

isok(n) = numdiv(eulerphi(n)) == sigma(factorback(factorint(n)[, 1])); \\ Michel Marcus, Jul 10 2019

{ n : A062821(n) = A048250(n) }.

Unspecific references removed by R. J. Mathar, Mar 26 2010

A176839 The number of iterations to reach 1 under the map x -> x-tau(phi(x)), starting at n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 5, 5, 5, 4, 6, 5, 6, 5, 7, 5, 7, 6, 7, 6, 8, 6, 7, 7, 7, 7, 9, 8, 8, 7, 8, 8, 10, 8, 9, 9, 11, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 10, 11, 11, 12, 11, 12, 12, 12, 12, 12, 12, 13, 11, 12, 13, 13, 12, 13, 13, 13, 12, 13, 14, 14, 14

Offset: 1

Michel Lagneau, Apr 27 2010

nonn

Tau(n) = A000005(n) is the number of divisors of n, and phi(n) = A000010(n) is the Euler totient function.

			a(12)=4 because
f(12) = 12 - tau(phi(12)) = 12 - tau(4) = 12 - 3 = 9;
f(9) = 9 - tau(phi(9)) = 9 - tau(6) = 9 - 4 = 5;
f(5) = 5 - tau(phi(5)) = 5 - tau(4) = 5 - 3 = 2;
f(2) = 2 - tau(phi(2)) = 2 - tau(1) = 2 - 1 = 1, and a(12) = 4.

Cf. A062821.

A062821 := proc(n)
        numtheory[tau](numtheory[phi](n)) ;
end proc:
A176839 := proc(n)
        a := 0 ;
        x := n ;
        while x <> 1 do
                x := x-A062821(x) ;
                a := a+1 ;
        end do:
        a ;
end proc: # R. J. Mathar, Oct 11 2011

f[n_] := If[n == 1, 1, n - DivisorSigma[0, EulerPhi[n]]];
a[n_] := Length[FixedPointList[f, n]] - 2;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 09 2024 *)

A216325 Number of divisors of the degree of the minimal polynomial for 2*cos(Pi/n), n >= 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 2, 4, 4, 6, 3, 6, 4, 4, 4, 5, 4, 5, 6, 6, 6, 6, 6, 5, 6, 6, 4, 6, 6, 4, 2, 5, 4, 6, 5, 8, 4, 6, 6, 8, 6, 6, 2, 5, 8, 8, 6, 6, 8, 6, 4, 6, 4, 8, 4, 8, 9, 9, 6, 9, 8, 8, 4, 6, 4, 8, 2, 8, 6, 8, 6, 8, 6, 8, 9, 6, 8, 4, 9, 6, 10, 8

Offset: 1

Wolfdieter Lang, Sep 27 2012

nonn

For the minimal polynomials C(n,x) of the algebraic number rho = 2*cos(Pi/n), n >= 1, see their coefficient table A187360. Their degree is delta(n)= phi(2*n)/2, if n >= 2, and delta(1) = 1, with Euler's totient A000010. The delta sequence is given in A055034. a(n) is the number of divisors of delta(n).
a(n) is also the number of distinct Modd n orders given in the table A216320 in row n. (For Modd n see a comment on A203571).
See the analog A062821(n), with the number of divisors of phi(n). The corresponding order table is A216327.

			a(8) = 3 because C(8,x) = x^4 - 4*x^2 + 2, with degree delta(8) = A055034(8) = 4, and the three divisors of 4 are 1, 2 and 4. tau(4) = A000005(4) = 3.

Cf. A062821 (analog).

a(n) = tau(delta(n)), n >= 1, with tau = A000005 (number of divisors), delta defined in a comment above and given as delta(n) = A055034(n).

A280338 Number of sizes of remainder sets for n, for any natural number c, given natural number b in (b^c) mod n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 3, 6, 4, 3, 3, 5, 4, 6, 5, 4, 4, 4, 4, 6, 6, 6, 6, 6, 3, 8, 6, 4, 5, 6, 6, 9, 6, 6, 6, 8, 4, 8, 7, 7, 4, 4, 5, 8, 6, 5, 9, 6, 6, 6, 8, 6, 6, 4, 5, 12, 8, 6, 7, 6, 4, 8, 9, 4, 6, 8, 8, 12, 9, 7, 10, 8, 6, 8, 6, 9, 8, 4, 6, 5, 8

Offset: 1

Jeptha Davenport, Dec 31 2016

nonn

			For a(1): b^c mod 1 = 0, so only 1 remainder set (0) is possible, and its size is 1.
For a(2): for any b, b^c will be even if b is even, or odd if b is odd, so b^c mod 2 has only 1 remainder for a given b (either (0), size 1, or (1), also size 1).
For a(5): choosing c for an arbitrary b, for b = 2, 2^2 mod 5 = 4, 2^3 mod 5 = 3, 2^4 mod 5 = 1, 2^5 mod 5 = 2, 2^6 mod 5 = 4, etc. (4 remainders); for base 4, 4^1 mod 5 = 4, 4^2 mod 5 = 1, 4^3 mod 5 = 4, etc. (2 remainders); for base 21, 21^1 mod 5 = 1, 21^819 mod 5 = 1, etc. (1 remainder); these are the only numbers of remainders which occur for any c given b for b^c modulo 5, so the number of remainder set sizes for n = 5 is 3 (4, 2, or 1-size remainder sets).
For a(100): number of remainder set sizes possible for any c given b is 10 (1, 2, 3, 4, 5, 6, 10, 11, 20, or 21-size remainder sets).

Jeptha Davenport, Table of n, a(n) for n = 1..500

First differs from A062821 at index n=15.

A300217 Numbers k such that tau(phi(k)) is a prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 17, 32, 34, 40, 48, 60, 85, 128, 136, 160, 170, 192, 204, 240, 1285, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 4080, 4369, 8192, 8224, 8704, 8738, 10240, 10280, 10880, 12288, 12336, 13056, 15360, 15420, 16320, 65537

Offset: 1

Jaroslav Krizek, Feb 28 2018

nonn

Numbers k such that A062821(k) = A000005(A000010(k)) is a prime.
Supersequence of A062514.
From Robert Israel, Mar 18 2018: (Start)
Numbers k such that A000010(k) = 2^(p-1) where p is prime.
Numbers of the form 2^m*Product_{i=1..k} (2^(2^(e_i))+1) where 2^(2^(e_i)+1) are distinct Fermat primes (A019434) and m + 1 + Sum_i 2^(e_i) is prime.  In particular the prime terms are A249759.
(End)
According to a comment in A009087, if the sum of divisors is prime, then the number of divisors is also prime. - Michael B. Porter, Mar 23 2018

			17 is a term because phi(17) = 16, tau(16) = 5 (prime).

Jaroslav Krizek, Table of n, a(n) for n = 1..100

Cf. A000005, A000010, A062514, A062821, A019434, A249759.

[n: n in[1..10^6] | IsPrime(NumberOfDivisors(EulerPhi(n)))];

select(isprime @ numtheory:-tau @ numtheory:-phi, [$1..10^5]); # Robert Israel, Mar 18 2018

Select[Range[2^16 + 1], PrimeQ@ DivisorSigma[0, EulerPhi@ #] &] (* Michael De Vlieger, Mar 01 2018 *)

isok(k) = isprime(numdiv(eulerphi(k))); \\ Altug Alkan, Mar 04 2018

cp's OEIS Frontend

A163374 a(n) = tau(tau(phi(n))).

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A163376 a(n) = phi(tau(phi(n))).

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A173334 Numbers k such that tau(phi(k)) = phi(sum-of-prime-divisors(k)).

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A173336 Numbers k such that tau(phi(k)) = sigma(sopf(k)).

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A173618 Numbers k such that tau(phi(k)) = rad(k).

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A173745 Numbers n such that tau(phi(n)) = sigma(rad(n)).

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