A173320 -id:A173320

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A173326

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

CROSSREFS

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

Cf. A173320, A062069, A001414, A001222.

KEYWORD

nonn

AUTHOR

Michel Lagneau, Feb 16 2010

EXTENSIONS

Corrected and edited by Michel Lagneau, Apr 25 2010

STATUS

approved

A173325

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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)

LINKS

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

FORMULA

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

EXAMPLE

k=3 with sigma(tau(3)) = sigma(2) = 3 = A008472(3).

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

MAPLE

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 :

CROSSREFS

Cf. A001222, A001414, A062069, A173320.

KEYWORD

nonn

AUTHOR

Michel Lagneau, Feb 16 2010

EXTENSIONS

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

STATUS

approved

A173617

Numbers n such that phi(tau(n))= rad(n)

+10
0

1, 4, 8, 32, 36, 192, 288, 768, 972, 1458, 5120, 13122, 326592, 19531250, 22588608, 46137344, 171532242

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

rad(n) is the product of the primes dividing n (A007947 ) tau(n) is the number of divisors of n (A000005) phi(n): Euler totient function (A000010)

a(18) > 10^10. - Donovan Johnson, Jul 27 2011

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..17.

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, Elementary theory of numbers, Warszawa, 1964.

Wikipedia, Euler's totient function

FORMULA

n such that A163109(n)= A007947(n)

EXAMPLE

tau(8) = 4, phi(4)=2 and rad(8)=2 tau(13122) = 18, phi(18)=6 and rad(13122)=6

MAPLE

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 :

CROSSREFS

Cf. A173320, A062069, A001414, A001222.

KEYWORD

nonn

AUTHOR

Michel Lagneau, Feb 22 2010

EXTENSIONS

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

STATUS

approved

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