cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Labos Elemer, Jan 25 2006

Keywords

Links

Crossrefs

Intersection of A018252 and A115558.
Cf. A000005, A000203, A076360, A076361, A062068, A062069, A115557, A115560.

Programs

  • Mathematica
    Select[Range[5000], !PrimeQ[#] && DivisorSigma[0, DivisorSigma[1, #^2]] == DivisorSigma[1, DivisorSigma[0, #^2]] &] (* Amiram Eldar, Jan 28 2025 *)
  • PARI
    isok(n)= !isprime(n) && (sigma(numdiv(n^2)) == numdiv(sigma(n^2))); \\ Michel Marcus, Dec 20 2013

Formula

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

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

Views

Author

Labos Elemer, Jan 25 2006

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    ds = DivisorSigma; Select[Range[1000]^2, ds[0, ds[1, #]] == ds[1, ds[0, #]] &] (* Giovanni Resta, Apr 29 2017 *)
  • PARI
    isok(n) = issquare(n) && (sigma(numdiv(n)) == numdiv(sigma(n))); \\ Michel Marcus, Dec 20 2013

Formula

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

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

Views

Author

Labos Elemer, Jan 25 2006

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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]
  • PARI
    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

Formula

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.

Extensions

More terms from Amiram Eldar, Jul 17 2019
Showing 1-3 of 3 results.

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