A062069 -id:A062069 - cp's OEIS frontend

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

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

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

A196225 Smallest number k such that sigma(tau(k)) = n, or 0 if there is no such k.

Original entry on oeis.org

1, 0, 2, 4, 0, 16, 6, 64, 0, 0, 0, 12, 36, 4096, 24, 0, 0, 48, 0, 262144, 0, 0, 0, 144, 0, 0, 0, 60, 0, 268435456, 120, 576, 0, 0, 0, 3072, 0, 68719476736, 180, 900, 0, 240, 0, 4398046511104, 0, 0, 0, 5184, 0, 0, 0, 0, 0, 196608, 0, 960, 46656, 0, 0, 360, 0, 1152921504606846976

Offset: 1

Jaroslav Krizek, Jan 02 2013

nonn

Smallest number k such that A062069(k) = A000203(A000005(k)) = n, or 0 if there is no such k.

			a(6) = 16 because number 16 is the smallest number k such that sigma(tau(k)) = 6; (tau(16) = 5, sigma(5) = 6).

Amiram Eldar, Table of n, a(n) for n = 1..3323
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).

Cf. A062069 (sigma(tau(n))), A000203(sigma(n)), A000005(tau(n)), A005179, A051444.

a(n) = {my(v = invsigma(n), kmin = 0); for(i = 1, #v, k = A005179(v[i]); if(kmin == 0 || k < kmin, kmin = k)); kmin;} \\ Amiram Eldar, Jan 21 2025, using Max Alekseyev's invphi.gp and R. J. Mathar's A005179(n)

a(n) = 0 iff A051444(n) = 0.

a(24) and a(48) corrected by Amiram Eldar, Jan 21 2025

A280583 a(n) = product of divisors of the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 8, 2, 8, 3, 8, 2, 36, 2, 8, 8, 5, 2, 36, 2, 36, 8, 8, 2, 64, 3, 8, 8, 36, 2, 64, 2, 36, 8, 8, 8, 27, 2, 8, 8, 64, 2, 64, 2, 36, 36, 8, 2, 100, 3, 36, 8, 36, 2, 64, 8, 64, 8, 8, 2, 1728, 2, 8, 36, 7, 8, 64, 2, 36, 8, 64, 2, 1728, 2, 8, 36, 36, 8

Offset: 1

Jaroslav Krizek, Jan 07 2017

nonn

			For n = 6; a(n) = product of divisors (tau(6)) = 1*2*4 = 8.

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

Cf. A000005, A000040, A001248, A007955, A010553, A062069.

[&*[d: d in Divisors(#[d: d in Divisors(n)])]: n in [1..100]]

Table[Times@@Divisors[DivisorSigma[0,n]],{n,80}] (* Harvey P. Dale, Dec 04 2021 *)

from math import isqrt
from sympy import divisor_count
def A280583(n): return (lambda m:(isqrt(m) if (c:=divisor_count(m)) & 1 else 1)*m**(c//2))(divisor_count(n)) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(A000005(n)).
a(p) = 2 for p = primes (A000040).
a(n) = 3 for squares of primes (A001248).

A113837 A number k is included if d(sigma(k)) > sigma(d(k)), where d(k) is number of positive divisors of k and sigma(k) is sum of positive divisors of k.

Original entry on oeis.org

5, 7, 11, 13, 14, 15, 17, 19, 22, 23, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 101, 102, 103, 104, 106, 107, 109, 110, 111, 113, 114

Offset: 1

Leroy Quet, Jan 23 2006

nonn

			d(sigma(14)) = d(24) = 8, and sigma(d(14)) = sigma(4) = 7. Since 8 > 7, 14 is in the sequence.

Cf. A062068, A062069.

with(numtheory): a:=proc(n) if tau(sigma(n))>sigma(tau(n)) then n else fi end: seq(a(n),n=1..122); # Emeric Deutsch, Feb 06 2006

okQ[n_] := DivisorSigma[0, DivisorSigma[1, n]] >
           DivisorSigma[1, DivisorSigma[0, n]];
Select[Range[120], okQ] (* Jean-François Alcover, Nov 14 2024 *)

More terms from Emeric Deutsch, Feb 06 2006

A336687 Numbers m such that tau(sigma(m)) and sigma(tau(m)) both divide m, where tau(m) is the number of divisors function (A000005) and sigma(m) is the sum of divisors function (A000203).

Original entry on oeis.org

1, 3, 4, 12, 64, 84, 140, 144, 162, 192, 336, 360, 420, 468, 480, 576, 600, 644, 720, 780, 1008, 1344, 1512, 1584, 1600, 1740, 1872, 2160, 2240, 2448, 2592, 2736, 2880, 2884, 3136, 3240, 3888, 4032, 4158, 4228, 4464, 4608, 4800, 5040, 5115, 5184, 5328, 5670, 6060, 6192, 6336

Offset: 1

Bernard Schott, Jul 31 2020

nonn

Conjecture: The only m such that m = tau(sigma(m))*sigma(tau(m)) are in {1,468,3240}. Verified for m up to 1*10^9. - Ivan N. Ianakiev, Aug 06 2020

			For 84: tau(84) = 12 and sigma(12) = 28 with 84/28 = 3. Also, sigma(84) = 224 and tau(224) = 12 with 84/12 = 7. Hence, 84 is a term.

Cf. A000005, A000203, A062068, A062069.

Intersection of A336612 and A336613.

with(numtheory):
filter:= m-> irem(m, tau(sigma(m)))=0 and irem(m, sigma(tau(m)))=0:
select(filter, [$1..7000])[];

Select[Range[6400], And @@ Divisible[#, {DivisorSigma[0, DivisorSigma[1, #]], DivisorSigma[1, DivisorSigma[0, #]]}] &] (* Amiram Eldar, Jul 31 2020 *)

isok(m) = !(m % numdiv(sigma(m))) && !(m % sigma(numdiv(m))); \\ Michel Marcus, Aug 02 2020

cp's OEIS Frontend

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

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

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A196225 Smallest number k such that sigma(tau(k)) = n, or 0 if there is no such k.

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A280583 a(n) = product of divisors of the number of divisors of n.

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A113837 A number k is included if d(sigma(k)) > sigma(d(k)), where d(k) is number of positive divisors of k and sigma(k) is sum of positive divisors of k.

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A336687 Numbers m such that tau(sigma(m)) and sigma(tau(m)) both divide m, where tau(m) is the number of divisors function (A000005) and sigma(m) is the sum of divisors function (A000203).

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