A173320 -id:A173320 - cp's OEIS frontend

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

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

A173325 Numbers k such that sigma(tau(k)) equals the sum of distinct primes dividing k.

Original entry on oeis.org

3, 10, 104, 105, 175, 245, 276, 343, 414, 484, 532, 798, 1190, 1430, 1776, 1862, 3105, 3174, 3712, 4394, 5049, 5054, 5104, 5994, 6256, 6360, 6975, 8125, 8480, 8625, 9472, 9648, 10600, 12408, 12789, 14310, 16544, 16625, 16728, 19908, 20295, 21056, 21708

Offset: 1

Michel Lagneau, Feb 16 2010

nonn

sigma(tau(k)) = A000203(A000005(k)) = A062069(k).
From Robert Israel, Nov 07 2016: (Start)
If m is in A023194, sigma(m)^(m-1) is in the sequence.
If p and q are distinct primes, and r and s are distinct primes such that r+s = (p+1)(q+1), then r^(p-1)*s^(q-1) is in the sequence.
(End)

			k=3 with sigma(tau(3)) = sigma(2) = 3 = A008472(3).
k=10 with sigma(tau(10)) = sigma(4) = 7 = A008472(10).

Robert Israel, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Glossary, Number of divisors
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

Cf. A001222, A001414, A062069, A173320.

with(numtheory): for n from 1 to 100000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if sigma(tau(n)) = t2 then print (n): else fi : od :

{k: A062069(k) = A008472(k)}.

"sopf" uses replaced and examples disentangled by R. J. Mathar, Feb 24 2010

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

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

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A173325 Numbers k such that sigma(tau(k)) equals the sum of distinct primes dividing k.

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

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