A163109 -id:A163109 - cp's OEIS frontend

A290085 a(n) = A289626(A000005(n)).

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 5, 2, 2, 2, 2, 1, 5, 1, 2, 2, 2, 2, 4, 1, 2, 2, 5, 1, 5, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 2, 4, 2, 5, 1, 2, 2, 5, 1, 5, 1, 2, 2, 2, 2, 5, 1, 3, 3, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 2, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 5, 5, 2, 1, 5, 1, 5, 2, 3, 1, 5, 2, 2, 2, 2, 2, 8

Offset: 1

Antti Karttunen, Aug 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A163109, A289626, A290086, A290087, A290088.

a(n) = A289626(A000005(n)).

A163378 a(n) = sigma(phi(tau(n))).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 7, 1, 3, 1, 3, 3, 3, 1, 7, 3, 3, 3, 3, 1, 7, 1, 3, 3, 3, 3, 12, 1, 3, 3, 7, 1, 7, 1, 3, 3, 3, 1, 7, 3, 3, 3, 3, 1, 7, 3, 7, 3, 3, 1, 7, 1, 3, 3, 12, 3, 7, 1, 3, 3, 7, 1, 7, 1, 3, 3, 3, 3, 7, 1, 7, 7, 3, 1, 7, 3, 3

Offset: 1

Jaroslav Krizek, Jul 25 2009

nonn

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

Cf. A000005, A000010, A000203, A062402, A163109.

Table[DivisorSigma[1,EulerPhi[DivisorSigma[0,n]]],{n,100}] (* Harvey P. Dale, Jun 21 2016 *)

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

a(n) = A000203(A000010(A000005(n))) = A000203(A163109(n)) = A062402(A000005(n)).

More terms from Harvey P. Dale, Jun 21 2016

A173326 Numbers k such that phi(tau(k)) = sopf(k).

Original entry on oeis.org

4, 8, 32, 1344, 2016, 2025, 2376, 3375, 3528, 4032, 4224, 4704, 4752, 5292, 5376, 5625, 6084, 6804, 7128, 9408, 9504, 10125, 10206, 10935, 12100, 12348, 12672, 16875, 16896, 20412, 21384, 23814, 26136, 28512, 29952, 30375, 31944, 32832, 42768, 46464, 48114

Offset: 1

Michel Lagneau, Feb 16 2010

nonn

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

Donovan Johnson, Table of n, a(n) for n = 1..1000
A. Bogomolny, Euler Function and Theorem
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
W. Sierpinski, Number Of Divisors And Their Sum

Cf. A000005 (tau), A000010 (phi), A008472 (sopf).

Cf. A173320, A062069, A001414, A001222.

A008472 := proc(n) add(p,p= numtheory[factorset](n)) ; end proc:
A163109 := proc(n) numtheory[phi](numtheory[tau](n)) ; end proc:
for n from 1 to 40000 do if A008472(n) = A163109(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Sep 02 2011

Select[Range[2,50000],EulerPhi[DivisorSigma[0,#]]==Total[ Transpose[ FactorInteger[#]][[1]]]&] (* Harvey P. Dale, Nov 15 2013 *)

{k: A163109(k) = A008472(k)}.

Corrected and edited by Michel Lagneau, Apr 25 2010

A385122 a(n) = d(phi(n)) - phi(d(n)) where d(n) = A000005(n) is the number of divisors and phi(n) = A000010(n) is the Euler totient function.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 1, 2, 1, 3, 1, 5, 2, 2, 0, 4, 2, 5, 2, 4, 2, 3, 0, 4, 4, 4, 4, 5, 0, 7, 3, 4, 3, 6, 0, 8, 4, 6, 1, 7, 2, 7, 4, 6, 2, 3, 1, 6, 4, 4, 6, 5, 2, 6, 4, 7, 4, 3, 1, 11, 6, 7, 0, 8, 2, 7, 4, 4, 4, 7, 4, 11, 7, 6, 7, 10, 4, 7, 2, 4, 6, 3, 4, 5, 6

Offset: 1

Sean A. Irvine, Jun 18 2025

sign

First negative value is a(120) = -2.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A062821, A078148, A078150, A163109.

A385122[n_] := DivisorSigma[0, EulerPhi[n]] - EulerPhi[DivisorSigma[0, n]];
Array[A385122, 100] (* Paolo Xausa, Jun 19 2025 *)

a(n) = numdiv(eulerphi(n)) - eulerphi(numdiv(n)); \\ Michel Marcus, Jun 19 2025

a(n) = A000005(A000010(n)) - A000010(A000005(n)).

a(n) = A062821(n) - A163109(n).

A163377 a(n) = tau(phi(tau(n))).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jul 25 2009

nonn

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

Cf. A000005, A000010, A062821, A163109.

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

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

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

A163379 a(n) = phi(phi(tau(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2

Offset: 1

Jaroslav Krizek, Jul 25 2009

nonn

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

Cf. A000005, A000010, A010554, A163109.

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

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

a(n) = A000010(A000010(A000005(n))) = A000010(A163109(n)) = A010554(A000005(n)).

More terms added by G. C. Greubel, Dec 20 2016

A163367 a(n) = phi(tau(sigma(n))).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 4, 4, 1, 2, 2, 2, 4, 2, 6, 4, 4, 1, 4, 4, 4, 4, 4, 2, 2, 4, 4, 4, 2, 2, 4, 4, 4, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 4, 2, 4, 8, 4, 8, 4, 4, 4, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 4, 4, 2, 4, 2, 4, 4, 8, 4, 4, 2, 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, A000203, A062068, A163109.

with(numtheory): A163367:=n->phi(tau(sigma(n))): seq(A163367(n), n=1..150); # Wesley Ivan Hurt, Dec 19 2016

Table[EulerPhi[DivisorSigma[0, DivisorSigma[1, n]]], {n, 100}] (* G. C. Greubel, Dec 19 2016 *)

vector(50, n, eulerphi(numdiv(sigma(n)))) \\ G. C. Greubel, Dec 19 2016

a(n) = A000010(A000005(A000203(n))) = A000010(A062068(n)) = A163109(A000203(n)).

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)).

A173328 Numbers k such that phi(tau(k)) = tau(sopf(k)).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 18, 20, 22, 25, 27, 30, 32, 34, 44, 49, 50, 58, 60, 68, 70, 82, 90, 102, 104, 105, 116, 118, 121, 125, 135, 140, 142, 150, 152, 164, 169, 174, 182, 189, 190, 195, 202, 204, 208, 214, 231, 236, 238, 242, 243, 246, 248, 252, 274, 284, 285, 286

Offset: 1

Michel Lagneau, Feb 16 2010

nonn

Sopf(n) = A008472(n) is the sum of the distinct primes dividing n, tau(n) = A000005(n) is the number of divisors of n, phi = A000010 is Euler's totient function.

			4 is in the sequence because tau(4) = 3, phi(3) = 2, sopf(4) = 2 and tau(2) = 2.
6 is in the sequence because tau(6) = 4, phi(6) = 2, sopf(6) = 5 and tau(5) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
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.

Cf. A000005 (tau), A000010 (phi), A008472 (sopfr), A163109.

isA173328 := proc(n)
        numtheory[phi](numtheory[tau](n)) = numtheory[tau](A008472(n)) ;
end proc:
for n from 1 to 300 do
        if isA173328(n) then
                printf("%d,",n);
        end if;
end do: # R. J. Mathar, Nov 07 2011

Select[Range[2,300],EulerPhi[DivisorSigma[0,#]]==DivisorSigma[0, Total[ FactorInteger[#][[All,1]]]]&] (* Harvey P. Dale, May 30 2017 *)

isok(k) = if(k == 1, 0, my(f=factor(k)); eulerphi(numdiv(f)) == numdiv(vecsum(f[,1]))); \\ Amiram Eldar, Feb 08 2025

{k : A163109(k) = tau(A008472(k))}.

A173617 Numbers k such that phi(tau(k)) = rad(k).

Original entry on oeis.org

1, 4, 8, 32, 36, 192, 288, 768, 972, 1458, 5120, 13122, 326592, 19531250, 22588608, 46137344, 171532242, 110000000000, 132799613957120, 1618481116086272, 506590324238281250, 8358680908399640576, 162805498773679522226642, 198263834416799184651812864, 7852841179377049820122874642432, 4299870835974154129876817272635392

Offset: 1

Michel Lagneau, Feb 22 2010

nonn

rad(n) is the product of the primes dividing n (A007947), tau(n) is the number of divisors of n (A000005), and phi(n) is Euler totient function (A000010).
Numbers k such that A163109(k) = A007947(k).
a(18) > 10^10. - Donovan Johnson, Jul 27 2011
From Amiram Eldar, Feb 08 2025: (Start)
1 is the only odd term in this sequence.
The number of terms with any given number of divisors is finite.
There are no terms whose number of divisors d equals 2 or in A049195, or when omega(phi(d)) > bigomega(d), where omega = A001221 and bigomega = A001222.
If p is a Sophie Germain prime (A005384), then 2*p^(2*p) is a term. (End)

			8 is a term since tau(8) = 4, phi(4) = 2 and rad(8) = 2.
13122 is a term  tau(13122) = 18, phi(18) = 6 and rad(13122) = 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, Mathematica code for A173617, 2025.
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 (tau), A000010 (phi), A007947 (rad).

Cf. A001221, A001222, A001414, A005384, A049195, A062069, A163109, A173320.

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

(* First program: see the links section. *)
(* Second program: *)
q[k_] := k == 1 || EvenQ[k] && Module[{f = FactorInteger[k]}, EulerPhi[Times @@ (f[[;;,2]] + 1)] == Times @@ f[[;;, 1]]]; Select[Range[400000], q] (* Amiram Eldar, Feb 08 2025 *)

isok(k) = if(k == 1, 1, if(k % 2, 0, my(f=factor(k)); eulerphi(numdiv(f)) == vecprod(f[,1]))); \\ Amiram Eldar, Feb 08 2025

a(14)-a(17) from Donovan Johnson, Jul 27 2011

a(18)-a(26) from Amiram Eldar, Feb 08 2025

cp's OEIS Frontend

A290085 a(n) = A289626(A000005(n)).

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A163378 a(n) = sigma(phi(tau(n))).

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A173326 Numbers k such that phi(tau(k)) = sopf(k).

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A385122 a(n) = d(phi(n)) - phi(d(n)) where d(n) = A000005(n) is the number of divisors and phi(n) = A000010(n) is the Euler totient function.

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A163377 a(n) = tau(phi(tau(n))).

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A163379 a(n) = phi(phi(tau(n))).

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A163367 a(n) = phi(tau(sigma(n))).

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

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