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 1-10 of 11 results. Next

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

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

Views

Author

Labos Elemer, Jan 25 2006

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[1000], DivisorSigma[0, DivisorSigma[1, #^2]] == DivisorSigma[1, DivisorSigma[0, #^2]] &] (* Amiram Eldar, Jan 28 2025 *)
  • PARI
    isok(k) = numdiv(sigma(k^2)) == sigma(numdiv(k^2)); \\ Amiram Eldar, Jan 28 2025

Formula

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

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.

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

A226117 Numbers n such that phi(sigma(tau(n))) = tau(sigma(phi(n))).

Original entry on oeis.org

1, 3, 4, 5, 14, 17, 20, 21, 22, 26, 51, 63, 65, 66, 72, 76, 80, 84, 90, 100, 106, 112, 132, 135, 150, 152, 165, 182, 190, 196, 221, 222, 232, 246, 255, 290, 291, 292, 294, 320, 326, 375, 386, 396, 424, 450, 460, 489, 530, 561, 567, 585, 588, 600, 606, 608, 615
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Examples

			For n=23529 we have:
phi(23529)=13200 -> sigma(13200)=46128 -> tau(46128)=30.
tau(23529)=16 -> sigma(16)=31 -> phi(31)=30.

Links

Crossrefs

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226118, A226119.

Programs

  • Maple
    with(numtheory); A226117:=proc(q) local n;
for n from 1 to q do
if phi(sigma(tau(n)))=tau(sigma(phi(n))) then print(n);
fi; od; end: A226117(10^6);
  • Mathematica
    Select[Range[700],EulerPhi[DivisorSigma[1,DivisorSigma[0,#]]] == DivisorSigma[ 0,DivisorSigma[ 1,EulerPhi[ #]]]&] (* Harvey P. Dale, Dec 12 2021 *)

A226118 Numbers n such that sigma(tau(phi(n))) = phi(tau(sigma(n))).

Original entry on oeis.org

1, 2, 136, 160, 170, 204, 240, 282, 716, 745, 1002, 1077, 1465, 1509, 1578, 1868, 2012, 2157, 2346, 2720, 2760, 3608, 3898, 4101, 4461, 4512, 5066, 5322, 5898, 6189, 7080, 7185, 7341, 7628, 7660, 8108, 8517, 8665, 8698, 8709, 8805, 8922, 8940, 9234, 9745, 9962
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Examples

			For n=9962 we have:
sigma(9962)=15876 -> tau(15876)=45 -> phi(45)=24.
phi(9962)=4672 -> tau(4672)=14 -> sigma(14)=24.

Links

Crossrefs

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226117, A226119.

Programs

  • Maple
    with(numtheory); A226118:=proc(q) local n;
for n from 1 to q do
if sigma(tau(phi(n)))=phi(tau(sigma(n))) then print(n);
fi; od; end: A226118(10^6);
  • Mathematica
    Select[Range[10000],EulerPhi[DivisorSigma[0,DivisorSigma[1,#]]] == DivisorSigma[ 1, DivisorSigma[ 0, EulerPhi[#]]]&] (* Harvey P. Dale, May 26 2016 *)

A226119 Numbers such that sigma(phi(tau(n)))=tau(phi(sigma(n))).

Original entry on oeis.org

1, 6, 36, 64, 105, 114, 135, 1980, 2016, 3072, 5120, 7056, 7840, 9216, 16320, 18720, 18900, 23100, 23622, 24003, 25536, 26088, 26733, 28455, 29078, 29337, 29700, 29760, 30597, 30894, 30912, 31155, 31496, 31758, 32361, 33782, 34020, 34286, 36000, 36036, 36099
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Examples

			29337 is in the sequence since:
sigma(29337)=49152 -> phi(49152)=16384 -> tau(16384)=15.
tau(29337)=16 -> phi(16)=8 -> sigma(8)=15.

Links

Crossrefs

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226117, A226118.

Programs

  • Maple
    with(numtheory); A226119:=proc(q) local n;
for n from 1 to q do
if sigma(phi(tau(n)))=tau(phi(sigma(n))) then print(n);
fi; od; end: A226119(10^6);
  • Mathematica
    Select[Range[36099], DivisorSigma[1, EulerPhi[DivisorSigma[0, #]]] == DivisorSigma[0, EulerPhi[DivisorSigma[1, #]]] &] (* T. D. Noe, May 28 2013 *)

A045835 Numbers n such that sopfr(Omega(n)) = Omega(sopfr(n)), where Omega(m) is the number and sopfr(m) is the sum of prime factors of m, with repetition respectively.

Original entry on oeis.org

1, 4, 9, 14, 18, 21, 25, 26, 33, 38, 42, 46, 49, 50, 57, 62, 69, 74, 78, 85, 92, 93, 94, 106, 110, 121, 129, 130, 133, 134, 138, 140, 145, 154, 164, 166, 169, 177, 178, 189, 204, 205, 213, 217, 218, 222, 225, 226, 230, 236, 237, 249, 253, 254, 262, 265, 266, 278
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 04 2003

Keywords

Comments

A001414(A001222(a(n))) = A001222(A001414(a(n)));
squares of primes (A001248) are a subsequence.

Examples

			n=189=3*3*3*7: Omega(sopfr(189))=Omega(3+3+3+7)=Omega(16)=4, sopfr(Omega(189))=sopfr(4)=2+2=4, therefore 189 is a term.

Crossrefs

Cf. A001222, A001414, A001248, A076361.

Programs

  • Mathematica
    sopfr[n_] := If[n == 1, 0, Total[Times @@@ FactorInteger[n]]];
Select[Range[1000], If[# == 1, True, sopfr[PrimeOmega[#]] == PrimeOmega[sopfr[#]]]&] (* Jean-François Alcover, Apr 06 2021 *)

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

Views

Author

Jaroslav Krizek, Jul 16 2009

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A000005, A000203, A076361, A076360.

Programs

Extensions

Edited by R. J. Mathar, Jul 21 2009

A218006 Numbers n such that sigma(tau(phi(n))) = tau(phi(sigma(n))) = phi(sigma(tau(n))).

Original entry on oeis.org

1, 34, 36, 96, 128, 468, 1200, 21216, 102060, 110976, 117684, 211428, 331380, 366660, 437220, 511680, 530712, 706560, 710388, 726240, 732240, 759360, 838080, 845376, 875840, 911040, 975936, 1014016, 1041216, 1093440, 1110720, 1141440, 1167696, 1289280
Offset: 1

Views

Author

Jayanta Basu, Mar 26 2013

Keywords

Comments

Here phi denotes Euler's totient function, tau(n) denotes number of divisors of n and sigma(n) denotes sum of all divisors of n. Only cyclic rotation of operators is considered.

Crossrefs

Cf. A000005, A000010, A000203, A033632, A076361, A078148.

Programs

  • Mathematica
    Select[Range[1000000], DivisorSigma[1, DivisorSigma[0, EulerPhi[#]]] == DivisorSigma[0, EulerPhi[DivisorSigma[1, #]]] == EulerPhi[DivisorSigma[1, DivisorSigma[0, #]]] &]
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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