A076360 -id:A076360 - 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

A076361 Numbers k such that d(sigma(k)) = sigma(d(k)).

Original entry on oeis.org

1, 3, 44, 49, 66, 68, 70, 76, 99, 121, 124, 147, 153, 164, 169, 170, 171, 172, 243, 245, 268, 275, 279, 361, 363, 387, 425, 475, 507, 529, 564, 603, 620, 644, 652, 724, 775, 841, 844, 845, 873, 891, 927, 948, 961, 964, 1075, 1083, 1132, 1324, 1348, 1377

Offset: 1

Labos Elemer, Oct 08 2002

Solutions to A076360(x) = 0.

Assuming Schinzel's hypothesis is true, the sequence is infinite. That conjecture implies that there are infinitely many primes p for which (p^2 + p + 1)/3 is prime. (E.g., p = 7, 13, 19, 31, 43, 73, 97, ...) For such p, we have d(sigma(p^2)) = d(p^2+p+1) = 4 and sigma(d(p^2)) = sigma(3) = 4, so p^2 is in the sequence. - Dean Hickerson, Jan 24 2006

Charles R Greathouse IV, Table of n, a(n) for n = 1..10029
Eric Weisstein's World of Mathematics, Schinzel's hypothesis.

Cf. A000005, A000203.

d0[x_] := DivisorSigma[0, x] d1[x_] := DivisorSigma[1, x] Do[s=d0[d1[n]]-d1[d0[n]]; If[s==0, Print[n]], {n, 1, 10000}]
Select[Range[1380],DivisorSigma[0, DivisorSigma[1, #]] == DivisorSigma[1, DivisorSigma[0, #]] &] (* Jayanta Basu, Mar 26 2013 *)

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

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

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

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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A076361 Numbers k such that d(sigma(k)) = sigma(d(k)).

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

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