A062069 -id:A062069 - cp's OEIS frontend

A062068 a(n) = d(sigma(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisor function (A000203).

1, 2, 3, 2, 4, 6, 4, 4, 2, 6, 6, 6, 4, 8, 8, 2, 6, 4, 6, 8, 6, 9, 8, 12, 2, 8, 8, 8, 8, 12, 6, 6, 10, 8, 10, 4, 4, 12, 8, 12, 8, 12, 6, 12, 8, 12, 10, 6, 4, 4, 12, 6, 8, 16, 12, 16, 10, 12, 12, 16, 4, 12, 8, 2, 12, 15, 6, 12, 12, 15, 12, 8, 4, 8, 6, 12, 12, 16, 10, 8, 3, 12, 12, 12, 12, 12

Offset: 1

Amarnath Murthy, Jun 13 2001

			a(5) = d(sigma(5)) = d(6) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A000203, A062068, A062069.

Table[DivisorSigma[0, DivisorSigma[1, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

PARI
```
for(n=1,120,print(numdiv(sigma(n))))
```

{ for (n=1, 1000, write("b062068.txt", n, " ", numdiv(sigma(n))) ) } \\ Harry J. Smith, Jul 31 2009

a(n) = A000005(A000203(n)) = A062069(n) + A076360(n). - Amiram Eldar, Mar 16 2025

Corrected and extended by Jason Earls, Jun 16 2001

A076360 a(n) = d(sigma(n)) - sigma(d(n)), where d(n) is the number of divisors of n and sigma(n) is their sum.

Original entry on oeis.org

0, -1, 0, -2, 1, -1, 1, -3, -2, -1, 3, -6, 1, 1, 1, -4, 3, -8, 3, -4, -1, 2, 5, -3, -2, 1, 1, -4, 5, -3, 3, -6, 3, 1, 3, -9, 1, 5, 1, -3, 5, -3, 3, 0, -4, 5, 7, -12, 0, -8, 5, -6, 5, 1, 5, 1, 3, 5, 9, -12, 1, 5, -4, -6, 5, 0, 3, 0, 5, 0, 9, -20, 1, 1, -6, 0, 5, 1, 7, -10, -3, 5, 9, -16, 5, 5, 9, 3, 9, -16, 3, 4, 1, 8, 9, -10, 3, -6, 0, -9, 5, 1, 5, 1, -1

Offset: 1

Labos Elemer, Oct 08 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A062068, A062069.

d0[x_] := DivisorSigma[0, x]
d1[x_] := DivisorSigma[1, x]
Table[ d0[d1[w]] - d1[d0[w]], {w, 1, 128}]

a(n) = {my(f = factor(n)); numdiv(sigma(f)) - sigma(numdiv(f));} \\ Amiram Eldar, Mar 16 2025

a(n) = A000005(A000203(n)) - A000203(A000005(n)) = A062068(n) - A062069(n). - Amiram Eldar, Mar 16 2025

A115557 Squares in A076361.

Original entry on oeis.org

1, 49, 121, 169, 361, 529, 841, 961, 1849, 2209, 2809, 5329, 6889, 9409, 10609, 12769, 16129, 24649, 32041, 38809, 39601, 49729, 51529, 54289, 57121, 58081, 63001, 66049, 73441, 78961, 96721, 99856, 100489, 110889, 124609, 151321, 160801

Offset: 1

Labos Elemer, Jan 25 2006

nonn

The commutator [sigma, tau] is zero, that is, A076360(x) = 0 and x is a square.

			The special squared prime 121 is a term because it is a square and sigma(tau(121)) = sigma(3) = 4 = tau(sigma(121)) = tau(1 + 11 + 121) = tau(133) = 4.
The least solution with composite square root is 316^2 = 99856: tau(99856) = 15, sigma(15) = 24 or sigma(99856) = 195951 = 3*7*7*31*43, tau(195951) = 24.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A062068, A062069, A076360, A076361, A115558.

ds = DivisorSigma; Select[Range[1000]^2, ds[0, ds[1, #]] == ds[1, ds[0, #]] &] (* Giovanni Resta, Apr 29 2017 *)

isok(n) = issquare(n) && (sigma(numdiv(n)) == numdiv(sigma(n))); \\ Michel Marcus, Dec 20 2013

a(n) = A115558(n)^2. - Amiram Eldar, Jan 31 2025

A115558 a(n) is the square root of A115557(n).

Original entry on oeis.org

1, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 73, 83, 97, 103, 113, 127, 157, 179, 197, 199, 223, 227, 233, 239, 241, 251, 257, 271, 281, 311, 316, 317, 333, 353, 389, 401, 409, 419, 421, 443, 449, 461, 467, 479, 491, 503, 509, 549, 563, 587, 593, 599, 617, 641

Offset: 1

Labos Elemer, Jan 25 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115559, A115560.

Select[Range[1000], DivisorSigma[0, DivisorSigma[1, #^2]] == DivisorSigma[1, DivisorSigma[0, #^2]] &] (* Amiram Eldar, Jan 28 2025 *)

isok(k) = numdiv(sigma(k^2)) == sigma(numdiv(k^2)); \\ Amiram Eldar, Jan 28 2025

The commutator [sigma, tau] is zero and a(n) is the square root of solutions. Both prime and composite numbers.

A115559 Nonprime terms of A115558.

Original entry on oeis.org

1, 316, 333, 549, 844, 963, 981, 1052, 1233, 1251, 1304, 1341, 1359, 1474, 1629, 1688, 1737, 1738, 1996, 2061, 2144, 2216, 2421, 2528, 2547, 2763, 2979, 3033, 3082, 3123, 3141, 3148, 3231, 3244, 3283, 3303, 3411, 3573, 3634, 3871, 3879, 3897, 3988, 4113

Offset: 1

Labos Elemer, Jan 25 2006

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000

Intersection of A018252 and A115558.

Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115560.

Select[Range[5000], !PrimeQ[#] && DivisorSigma[0, DivisorSigma[1, #^2]] == DivisorSigma[1, DivisorSigma[0, #^2]] &] (* Amiram Eldar, Jan 28 2025 *)

isok(n)= !isprime(n) && (sigma(numdiv(n^2)) == numdiv(sigma(n^2))); \\ Michel Marcus, Dec 20 2013

The commutator [sigma, tau] is zero and a(n) is the square root of solutions. Moreover this square root is a nonprime number.

A115560 Twin prime pairs k-1 and k+1 such that the squares of both are present in A115557.

Original entry on oeis.org

11, 13, 29, 31, 197, 199, 239, 241, 419, 421, 659, 661, 881, 883, 1019, 1021, 1061, 1063, 1481, 1483, 1877, 1879, 3167, 3169, 3821, 3823, 4019, 4021, 4049, 4051, 4787, 4789, 6359, 6361, 7589, 7591, 9437, 9439, 13691, 13693, 14447, 14449, 14627, 14629, 16451, 16453

Offset: 1

Labos Elemer, Jan 25 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115558, A115559.

ta={{0}};tb={{0}}; Do[s=DivisorSigma[1,DivisorSigma[0,n]]; s1=DivisorSigma[0,DivisorSigma[1,n]]; If[Equal[s-s1,0]&&IntegerQ[Sqrt[n]&&PrimeQ[Sqrt[n]]],Print[n]; ta=Append[ta,n];tb=Append[tb,Sqrt[n]]],{n,1,100000000}] ta=Delete[ta,1];tb=Delete[tb,1];ni=Intersection[tb,2+tb]; Union[ni,ni-2]

isok(n) = issquare(n) && (sigma(numdiv(n)) == numdiv(sigma(n))); \\ A115557
lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isok(p^2) && isok((p+2)^2), print1(p, ", ", p+2, ", ")););} \\ Michel Marcus, Jul 17 2019

The commutator [sigma, tau] is zero and a(n) is the square root of special prime solutions. These solutions are twin primes. Both twins are displayed.

More terms from Amiram Eldar, Jul 17 2019

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

A193349 Sum of odd divisors of tau(n).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 6, 1, 4, 1, 4, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 6, 4, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 8, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 1, 6, 6, 1, 1, 4, 1, 1

Offset: 1

Michel Lagneau, Jul 23 2011

nonn

			a(36) = 13 because tau(36) = 9 and the sum of the 3 odd divisors  {1, 3, 9} is 13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000593, A036537, A062069, A193348.

Table[Total[Select[Divisors[DivisorSigma[0,n]], OddQ[ # ]&]], {n, 80}]

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

a(n) = A000593(A000005(n)). - Reinhard Zumkeller, Jul 25 2011
From Amiram Eldar, Aug 12 2024: (Start)
a(n) = 1 if and only if n is in A036537.
a(n) = A062069(n) if and only if n is a square. (End)

A163106 a(n) = sigma(sigma(tau(n))), where tau = number of divisors and sigma = sum of divisors.

Original entry on oeis.org

1, 4, 4, 7, 4, 8, 4, 8, 7, 8, 4, 28, 4, 8, 8, 12, 4, 28, 4, 28, 8, 8, 4, 24, 7, 8, 8, 28, 4, 24, 4, 28, 8, 8, 8, 14, 4, 8, 8, 24, 4, 24, 4, 28, 28, 8, 4, 39, 7, 28, 8, 28, 4, 24, 8, 24, 8, 8, 4, 56, 4, 8, 28, 15, 8, 24, 4, 28, 8, 24, 4, 56, 4, 8, 28, 28, 8, 24, 4, 39, 12, 8, 4, 56, 8, 8, 8, 24, 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

Cf. A000005, A000203, A051027, A062069.

with(numtheory) : A163106 := proc(n) sigma(sigma(tau(n))) ; end: seq(A163106(n),n=1..120) ; # R. J. Mathar, Jul 27 2009

DivisorSigma[1,DivisorSigma[1,DivisorSigma[0,Range[90]]]] (* Harvey P. Dale, Oct 14 2017 *)

A163106(n) = sigma(sigma(numdiv(n))); \\ Antti Karttunen, May 25 2017

a(n) = A000203(A000203(A000005(n))) = A000203(A062069(n)) = A051027(A000005(n)).

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

A163375 a(n) = sigma(tau(phi(n))).

Original entry on oeis.org

1, 1, 3, 3, 4, 3, 7, 4, 7, 4, 7, 4, 12, 7, 7, 7, 6, 7, 12, 7, 12, 7, 7, 7, 12, 12, 12, 12, 12, 7, 15, 6, 12, 6, 15, 12, 13, 12, 15, 6, 15, 12, 15, 12, 15, 7, 7, 6, 15, 12, 12, 15, 12, 12, 15, 15, 13, 12, 7, 6, 28, 15, 13, 12, 18, 12, 15, 12, 12, 15, 15, 15

Offset: 1

Jaroslav Krizek, Jul 25 2009

nonn

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

Cf. A000005, A000010, A000203, A062069, A062821.

[SumOfDivisors(NumberOfDivisors(EulerPhi(n))): n in [1..80]]; // Vincenzo Librandi, Dec 20 2016

DivisorSigma[1, DivisorSigma[0, EulerPhi[Range[100]]]] (* G. C. Greubel, Dec 20 2016 *)

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

a(n) = A000203(A000005(A000010(n))) = A000203(A062821(n)) = A062069(A000010(n)).

cp's OEIS Frontend

A062068 a(n) = d(sigma(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisor function (A000203).

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A076360 a(n) = d(sigma(n)) - sigma(d(n)), where d(n) is the number of divisors of n and sigma(n) is their sum.

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A115557 Squares in A076361.

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A115558 a(n) is the square root of A115557(n).

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A115559 Nonprime terms of A115558.

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A115560 Twin prime pairs k-1 and k+1 such that the squares of both are present in A115557.

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Mathematica

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

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A193349 Sum of odd divisors of tau(n).