A062068 -id:A062068 - cp's OEIS frontend

A062069 a(n) = sigma(d(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisors function (A000203).

1, 3, 3, 4, 3, 7, 3, 7, 4, 7, 3, 12, 3, 7, 7, 6, 3, 12, 3, 12, 7, 7, 3, 15, 4, 7, 7, 12, 3, 15, 3, 12, 7, 7, 7, 13, 3, 7, 7, 15, 3, 15, 3, 12, 12, 7, 3, 18, 4, 12, 7, 12, 3, 15, 7, 15, 7, 7, 3, 28, 3, 7, 12, 8, 7, 15, 3, 12, 7, 15, 3, 28, 3, 7, 12, 12, 7, 15, 3, 18, 6, 7, 3, 28, 7, 7, 7, 15, 3

Offset: 1

Amarnath Murthy, Jun 13 2001

nonn

a(1) = 1, a(p) = 3 for p = primes (A000040), a(pq) = 7 for pq = product of two distinct primes (A006881), a(pq...z) = 2^(k+1)-1 = A000225(k+1) for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = sigma(k+1) = A000203(k+1) for p^k = prime powers (A000961(n) for n > 1). Sequence {1,3,4,12} is finite sequence of numbers n such that sigma(tau(n)) = n. [Jaroslav Krizek, Jul 16 2009]
For semiprime n, a(n) is either 4 or 7. Also a(n) = d(n) + omega(n) + mu(n), the sum of three core sequences A000005, A001221 and A008683. When n is semiprime, a(n) is completely defined by the Mobius function as: a(n) = 4 + 3*mu(n). a(n) also has the fractal-like identities a(d(n)) = d(n) and a(n) = sigma(a(d(n))). - Wesley Ivan Hurt, Sep 02 2013
If n is a triprime (A014612), d(n) is 4, 6, or 8 and a(n) = sigma(d(n)) is 7, 12, or 15 respectively. Then a(n) = -d(n)^2/4 + 5*d(n) - 9. - Wesley Ivan Hurt, Sep 08 2013

			sigma(d(12)) = sigma(6) = 12.

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

Cf. A000005, A000203, A001221, A008683, A062068.

A062069:= (n-> numtheory[sigma](numtheory[tau](n))):
seq (A062069(n), n=1..40); # Jani Melik, Jan 25 2011

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

v=[]; for(n=1,150,v=concat(v, sigma(numdiv(n)))); v

{ for (n=1, 1000, write("b062069.txt", n, " ", sigma(numdiv(n))) ) } \\ Harry J. Smith, Jul 31 2009

a(n) = A000203(A000005(n)). - Wesley Ivan Hurt, Sep 09 2013

More terms from Jason Earls, Jun 19 2001

A073802 Number of common divisors of n and sigma(n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 13 2002

nonn

From Jaroslav Krizek, Feb 18 2010: (Start)
Number of divisors d of number n such that d divides sigma(n).
a(n) = A000005(n) - A173438(n).
a(n) = A000005(n) for multiply-perfect numbers (A007691). (End)

			For n = 12: a(12) = 3; sigma(12) = 28, divisors of 12: 1, 2, 3, 4, 6, 12; d divides sigma(n) for 3 divisors d: 1, 2, 4.
n=96: d(96) = {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96}, d(sigma(96)) = {1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252}, CD(n, sigma(n)) = {1, 2, 3, 4, 6, 12} so a(96) = 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A062068, A073803, A073804.

[NumberOfDivisors(GCD(SumOfDivisors(n),n)): n in [1..100]]; // Vincenzo Librandi, Oct 09 2017

g1[x_] := Divisors[x]; g2[x_] := Divisors[DivisorSigma[1, x]]; ncd[x_] := Length[Intersection[g1[x], g2[x]]]; Table[ncd[w], {w, 1, 128}]
Table[Length[Intersection[Divisors[n], Divisors[DivisorSigma[1, n]]]], {n, 100}] (* Vincenzo Librandi, Oct 09 2017 *)
a[n_] := DivisorSigma[0, GCD[n, DivisorSigma[1, n]]]; Array[a, 100] (* Amiram Eldar, Nov 21 2024 *)

a(n)=numdiv(gcd(sigma(n),n)) \\ Charles R Greathouse IV, Mar 09 2014

See program.

a(n) = A000005(A009194(n)) = tau(gcd(n,sigma(n))). [Reinhard Zumkeller, Mar 12 2010]

A037197 Numbers k such that tau(sigma(k)) = tau(k) where tau(k) is the number of divisors of k and sigma(k) their sum.

Original entry on oeis.org

1, 2, 8, 12, 32, 52, 75, 84, 90, 98, 128, 150, 156, 338, 360, 392, 525, 528, 560, 600, 722, 867, 912, 972, 1050, 1352, 1452, 1456, 1525, 1734, 1922, 2064, 2160, 2340, 2400, 2888, 2890, 3050, 3120, 3216, 3698, 3744, 3872, 4080, 4144, 4200, 4500, 4575

Offset: 1

Naohiro Nomoto

nonn

			k = 75: divisors(75) = {1, 3, 5, 15, 25, 75}, divisors(sigma(75)) = divisors(124) = {1, 2, 4, 31, 62, 124}, both 75 and sigma(75) have 6 divisors, so 75 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A000005, A000203, A062068, A073802, A073803, A073804.

Do[s=DivisorSigma[0, DivisorSigma[1, n]]; s0=DivisorSigma[0, n]; If[Greater[s0, s], Print[n]], {n, 1, 1000}]
Select[Range[4600],DivisorSigma[0,#]==DivisorSigma[0,DivisorSigma[1,#]]&] (* Harvey P. Dale, Feb 08 2025 *)

is(n)=numdiv(sigma(n))==numdiv(n) \\ Charles R Greathouse IV, Feb 13 2013

Solutions to A000005(x) = A062068(x) = A000005(A000203(x)).

Conjecture: for n > 10^6, a(n) < n^2. - Benoit Cloitre, Aug 24 2002

Offset corrected by Reinhard Zumkeller, Jun 18 2009

Name edited by Michel Marcus, Nov 12 2023

A073803 Numbers k such that the number of divisors of k is smaller than that of sigma(k).

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

Labos Elemer, Aug 13 2002

nonn

			k = 96: divisors(96) = {1,2,3,4,6,8,12,16,24,32,48,96}, 12 divisors; divisors(sigma(96)) = {1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252}, 18 divisors; 12 < 18, so 96 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A062068, A073802, A073804, A037197.

filter:= proc(n) uses numtheory; tau(n) < tau(sigma(n)) end proc:
select(filter, [$1..100]); # Robert Israel, Aug 03 2020

Do[s=DivisorSigma[0, DivisorSigma[1, n]]; s0=DivisorSigma[0, n]; If[Greater[s0, s], Print[n]], {n, 1, 1000}]
Select[Range[100],DivisorSigma[0,#]Harvey P. Dale, Sep 22 2019 *)

isok(k) = {my(f = factor(k)); numdiv(f) < numdiv(sigma(f));} \\ Amiram Eldar, Mar 07 2025

Solutions to A000005(x) < A062068(x) = A000005(A000203(x)).

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

A073804 Numbers k such that the number of divisors of k is greater than that of sigma(k).

Original entry on oeis.org

4, 9, 16, 18, 25, 36, 48, 50, 64, 72, 80, 81, 100, 112, 144, 162, 180, 192, 196, 200, 208, 225, 240, 252, 256, 288, 289, 300, 320, 324, 336, 400, 432, 441, 448, 450, 468, 484, 512, 576, 578, 592, 624, 625, 648, 676, 700, 704, 720, 729, 768, 784, 800, 810, 832

Offset: 1

Labos Elemer, Aug 13 2002

nonn

			k = 25: divisors(25) = {1,5,25}, 3 divisors; divisors(sigma(25)) = {1,31}, 2 divisors; 2 < 3, so 25 is a term.
k = 48: divisors(48) = {1,2,3,4,6,8,12,16,24,48}, 10 divisors; divisors(sigma(48)) = {1,2,4,31,62,124}, 6 divisors, 6 < 10 so 48 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A000203, A062068, A073802, A073803, A037197.

Do[s=DivisorSigma[0, DivisorSigma[1, n]]; s0=DivisorSigma[0, n]; If[Greater[s0, s], Print[n]], {n, 1, 1000}]
Select[Range[900],DivisorSigma[0,#]>DivisorSigma[0,DivisorSigma[1,#]]&] (* Harvey P. Dale, Jan 18 2017 *)

isok(k) = {my(f = factor(k)); numdiv(f) > numdiv(sigma(f));} \\ Amiram Eldar, Mar 07 2025

Solutions to A000005(x) > A062068(x) = A000005(A000203(x)).

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.

A173320 Numbers k such that tau(sigma(k)) = sopf(k).

Original entry on oeis.org

2, 3, 4, 15, 16, 42, 45, 64, 81, 84, 245, 336, 340, 342, 460, 539, 550, 580, 605, 684, 882, 1012, 1014, 1160, 1344, 1360, 1640, 1674, 1700, 1785, 1840, 1972, 2178, 2254, 2320, 2322, 2736, 3096, 3348, 3645, 4048, 4096, 4212, 4332, 4389, 4400, 4644, 4830, 5022

Offset: 1

Michel Lagneau, Feb 16 2010

nonn

sopf(k) is the sum of the distinct primes dividing k (A008472), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of divisor of k (A000203).

			sigma(2) = 3, tau(3) = 2 and sopf(2) = 2 sigma(2254) = 4104, tau(4104) = 32 and sopf(2254) = 32.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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 (1919), 75-113.
Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Cf. A000005, A000203, A001414 (sopfr), A001222.

with(numtheory): for n from 1 to 12000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)): if tau(sigma(n)) = t2 then print (n): else fi : od :

Select[Range[2,5100],DivisorSigma[0,DivisorSigma[1,#]]==Total[ FactorInteger[ #][[All,1]]]&] (* Harvey P. Dale, May 31 2019 *)

k such that A062068(k)= A008472(k).

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.

cp's OEIS Frontend

A062069 a(n) = sigma(d(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisors function (A000203).

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A073802 Number of common divisors of n and sigma(n).

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A037197 Numbers k such that tau(sigma(k)) = tau(k) where tau(k) is the number of divisors of k and sigma(k) their sum.

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A073803 Numbers k such that the number of divisors of k is smaller than that of sigma(k).

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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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A073804 Numbers k such that the number of divisors of k is greater than that of sigma(k).

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A115557 Squares in A076361.

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