A062069 -id:A062069 - cp's OEIS frontend

A173327 Numbers k such that tau(phi(k))= sopf(k).

4, 45, 48, 75, 160, 180, 252, 294, 300, 315, 336, 351, 378, 396, 475, 507, 560, 605, 616, 650, 833, 936, 1216, 1375, 1452, 1690, 1805, 1920, 2023, 2112, 2200, 2349, 2496, 2736, 3211, 3520, 3648, 4095, 4160, 4256, 4332, 4389, 4464, 4477, 4508, 4620, 4693

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); and sopf(k) is the sum of the distinct primes dividing k without repetition (A008472).

			4 is in the sequence because phi(4) = 2, tau(2)=2 and sopf(4)=2 ;
45 is in the sequence because phi(45) = 24, tau(24)=8 and sopf(45)=8.

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
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.
Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.
Wikipedia, Euler's totient function

See A173326, A062069. Cf. A001414, A008472(sopfr), A001222, A062821.

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

for n from 1 to 150000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = t2 then print (n): else fi : od :

tpsQ[n_]:=DivisorSigma[0,EulerPhi[n]]==Total[Transpose[FactorInteger[n]][[1]]]; Select[Range[5000],tpsQ] (* Harvey P. Dale, Apr 10 2013 *)

sopf(n)=my(f=factor(n)[1,]);sum(i=1,#f,f[i])
is(n)=numdiv(eulerphi(n))==sopf(n) \\ Charles R Greathouse IV, May 20 2013

Numbers n such that A062821(n)= A008472(n)

Added punctuation to the examples. Corrected and edited by Michel Lagneau, Apr 25 2010

Edited by D. S. McNeil, Nov 20 2010

A181183 a(n) = sigma(tau(n)) (mod 2).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1

Offset: 1

Jani Melik, Jan 26 2011

nonn

Cf. A000035, A062069.

A181183:= n-> (numtheory[sigma](numtheory[tau](n)) mod 2):
seq (A181183(n), n=1..105);

Array[Mod[DivisorSigma[1, DivisorSigma[0, #]], 2] &, 105] (* Michael De Vlieger, Nov 18 2017 *)

A181183(n) = (sigma(numdiv(n))%2); \\ Antti Karttunen, Nov 18 2017

a(n) = A000035(A062069(n)). - Antti Karttunen, Nov 18 2017

A193350 Sum of even divisors of tau(n).

Original entry on oeis.org

0, 2, 2, 0, 2, 6, 2, 6, 0, 6, 2, 8, 2, 6, 6, 0, 2, 8, 2, 8, 6, 6, 2, 14, 0, 6, 6, 8, 2, 14, 2, 8, 6, 6, 6, 0, 2, 6, 6, 14, 2, 14, 2, 8, 8, 6, 2, 12, 0, 8, 6, 8, 2, 14, 6, 14, 6, 6, 2, 24, 2, 6, 8, 0, 6, 14, 2, 8, 6, 14, 2, 24, 2, 6, 8, 8, 6, 14, 2, 12, 0, 6, 2, 24, 6, 6, 6, 14, 2, 24, 6, 8, 6, 6, 6, 24, 2, 8, 8, 0

Offset: 1

Michel Lagneau, Jul 23 2011

nonn

			a(24) = 14 because tau(24) = 8 and the sum of the 3 even divisors {2, 4, 8} is 14.

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

Cf. A000005, A062069, A146076, A193347, A193349.

Cf. A000290 (the positions of zeros).

Table[Total[Select[Divisors[DivisorSigma[0,n]], EvenQ[ # ]&]], {n, 74}]

PARI
```
a(n)=sumdiv(sigma(n,0),d,(1-d%2)*d);
```

a(n) = A146076(A000005(n)). - Antti Karttunen, May 28 2017

a(n) = A062069(n) - A193349(n). - Amiram Eldar, Jan 27 2025

Data section extended to 100 terms by Antti Karttunen, May 28 2017

A336612 Numbers m such that sigma(tau(m)) divides 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, 14, 21, 30, 35, 64, 77, 84, 91, 105, 119, 133, 135, 140, 144, 161, 162, 165, 192, 195, 203, 217, 224, 255, 259, 285, 287, 301, 308, 329, 336, 343, 345, 360, 364, 371, 375, 392, 413, 420, 427, 435, 465, 468, 469, 476, 480, 497, 511, 532, 540, 553, 555, 576

Offset: 1

Bernard Schott, Jul 27 2020

nonn

Every 7*p with p prime <> 7 is a term because 7*p / sigma(tau(7*p)) = p (see example).

			35 = 7 * 5, tau(35) = 4, sigma(tau(35)) = sigma(4) = 4 + 2 + 1 = 7 and 35/7 = 5 hence 35 is a term.

Cf. A000005, A000203, A062069.

Cf. A336613 (tau(sigma(m)) divides m).

with(numtheory) filter:= m -> m/sigma(tau(m)) = floor(m/sigma(tau(m))) : select(filter, [$1..600]);

Select[Range[600], Divisible[#, DivisorSigma[1, DivisorSigma[0, #]]] &] (* Amiram Eldar, Jul 27 2020 *)

isok(m) = !(m % sigma(numdiv(m))); \\ Michel Marcus, Jul 29 2020

A162880 Numbers k such that tau(sigma(k)) is not equal to sigma(tau(k)).

Original entry on oeis.org

2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80

Offset: 1

Jaroslav Krizek, Jul 16 2009

nonn

The complement of A076361, that is, the indices k where A076360(k) is not zero.

			a(6)= 8 is in the list because tau(sigma(8))=A062068(8)=4 whereas sigma(tau(8)) = A062069(8) = 7.

Cf. A000005, A000203, A076361, A076360.

is(n)=numdiv(sigma(n))!=sigma(numdiv(n)) \\ Charles R Greathouse IV, Jun 25 2013

Edited by R. J. Mathar, Jul 21 2009

A163105 a(n) = tau(sigma(tau(n))), where tau = number of divisors of n (A000005), and sigma = sum of divisors of n (A000203).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jul 20 2009

nonn

Repeated application of tau (number of divisors) and sigma (sum of divisors).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000203, A062068, A062069, A163106.

with(numtheory) : A163105 := proc(n) tau(sigma(tau(n))) ; end: seq(A163105(n),n=1..120) ; # R. J. Mathar, Jul 27 2009

With[{s = DivisorSigma}, s[0, s[1, s[0, Range[100]]]]] (* Paolo Xausa, Oct 07 2024 *)

A163105(n) = numdiv(sigma(numdiv(n))); \\ Antti Karttunen, Jul 23 2017

a(n) = A000005(A000203(A000005(n))) = A000005(A062069(n)) = A062068(A000005(n)).

More terms from R. J. Mathar, Jul 27 2009

Name edited by Antti Karttunen, Jul 23 2017

A163108 a(n) = sigma(tau(sigma(n))).

Original entry on oeis.org

1, 3, 4, 3, 7, 12, 7, 7, 3, 12, 12, 12, 7, 15, 15, 3, 12, 7, 12, 15, 12, 13, 15, 28, 3, 15, 15, 15, 15, 28, 12, 12, 18, 15, 18, 7, 7, 28, 15, 28, 15, 28, 12, 28, 15, 28, 18, 12, 7, 7, 28, 12, 15, 31, 28, 31, 18, 28, 28, 31, 7, 28, 15, 3, 28, 24, 12, 28, 28, 24, 28, 15, 7, 15, 12, 28, 28

Offset: 1

Jaroslav Krizek, Jul 20 2009

nonn

Repeated application of tau (number of divisors) and sigma (sum of divisors).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A062068, A062069, A163107.

with(numtheory) : A163108 := proc(n) sigma(tau(sigma(n))) ; end: seq(A163108(n),n=1..120) ; # R. J. Mathar, Jul 27 2009

A163108(n) = sigma(numdiv(sigma(n))); \\ Antti Karttunen, Nov 18 2017

a(n) = A000203(A000005(A000203(n))) = A000203(A062068(n)) = A062069(A000203(n)).

More terms from R. J. Mathar, Jul 27 2009

A163368 a(n) = phi(sigma(tau(n))).

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 6, 2, 6, 2, 4, 2, 6, 6, 2, 2, 4, 2, 4, 6, 6, 2, 8, 2, 6, 6, 4, 2, 8, 2, 4, 6, 6, 6, 12, 2, 6, 6, 8, 2, 8, 2, 4, 4, 6, 2, 6, 2, 4, 6, 4, 2, 8, 6, 8, 6, 6, 2, 12, 2, 6, 4, 4, 6, 8, 2, 4, 6, 8, 2, 12, 2, 6, 4, 4, 6, 8, 2, 6, 2, 6, 2, 12, 6

Offset: 1

Jaroslav Krizek, Jul 25 2009

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

Cf. A000005, A000010, A000040, A000203, A006881, A053287, A062069, A062401, A120944.

[EulerPhi(SumOfDivisors(NumberOfDivisors(n))): n in [1..80]]; // Vincenzo Librandi, Dec 21 2016

with(numtheory): A163368:=n->phi(sigma(tau(n))): seq(A163368(n), n=1..150); # Wesley Ivan Hurt, Dec 19 2016

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

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

a(n) = A000010(A000203(A000005(n))) = A000010(A062069(n)) = A062401(A000005(n)).

a(1) = 1, a(p) = 2 for p = primes (A000040), a(pq) = 6 for pq = product of two distinct primes (A006881), a(pq...z) = A000010(2^(k+1)-1) = A053287(k+1) for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944).

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

A173582 Numbers k such that sigma(tau(k)) = rad(k).

Original entry on oeis.org

1, 3, 135, 336, 343, 375, 1134, 14406, 24336, 41067, 54756, 85293, 321408, 428544, 430080, 1028196, 1084752, 1651104, 1886976, 2476656, 2935296, 3066336, 3341637, 3577392, 4599504, 4881384, 5133375, 5366088, 5451264, 8347248, 8989344, 9240075, 9552816, 9871875

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), sigma(k) is the sum of divisor of k (A000203).

			tau(3) = 2, sigma(2) = 3 and rad(3) = 3. tau(135) = 8, sigma(8) = 15 and rad(135) = 15. tau(14406) = 20, sigma(20) = 42 and rad(14406) = 42.

Amiram Eldar, Table of n, a(n) for n = 1..100
C. K. Caldwell, The Prime Glossary, Number of divisors
Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Cf. A000005, A000203, A007947, A062069.

with(numtheory):for n from 1 to 1000000 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 :

Select[Range[500000], DivisorSigma[1, DivisorSigma[0, #]] == Times @@ (First@# & /@ FactorInteger[#]) &] (* Amiram Eldar, Jul 11 2019 *)

k such that A062069(k) = A007947(k).

a(20)-a(34) from Donovan Johnson, Jan 14 2012

cp's OEIS Frontend

A173327 Numbers k such that tau(phi(k))= sopf(k).

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A181183 a(n) = sigma(tau(n)) (mod 2).

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A193350 Sum of even divisors of tau(n).

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A336612 Numbers m such that sigma(tau(m)) divides 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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A162880 Numbers k such that tau(sigma(k)) is not equal to sigma(tau(k)).

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A163105 a(n) = tau(sigma(tau(n))), where tau = number of divisors of n (A000005), and sigma = sum of divisors of n (A000203).

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A163108 a(n) = sigma(tau(sigma(n))).

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