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.

A327056 RMS numbers (A140480) that are not arithmetic (A003601).

Original entry on oeis.org

2217231104, 6221622528, 9644780288, 12127073024, 15377570560, 15520617728, 22426778880, 43551357696, 67513462016, 84107119360, 84889511168, 90906475264, 107642993920, 156987452160, 255086523648, 446676800768, 497209993984, 529918233856, 588749835520, 636345326848
Offset: 1

Views

Author

Jaroslav Krizek, Oct 18 2019

Keywords

Comments

Numbers m such that the quadratic mean (the root mean square) of the divisors of m is an integer but the arithmetic mean of the divisors of m is not an integer.
Numbers m such that Q(m) = sqrt(A001157(m) / A000005(m)) is an integer but A(m) = A000203(m) / A000005(m) is not an integer.
Corresponding values of Q(m): 247511537, 368213825, 763370125, 957355945, 1237557685, 1237557685, 957355945, 1841069125, ...
Corresponding values of A(m): 418652080/9, 433603940/9, 324455362/3, 1166788784/9, 575646610/3, 1674608320/9, 315348320/3, ...
Complement of A327055 with respect to A140480.
Up to 10^13 there is only one odd term, a(29) = 3486482785825. Note that among the 7430 RMS numbers below 10^13 only 83 are even. - Giovanni Resta, Oct 29 2019

Links

Crossrefs

Cf.: A000005, A000203, A001157, A003601, A140480, A327055.

Programs

  • Magma
    [m: m in [1..10^6] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(Sqrt(&+[d^2: d in Divisors(m)] / NumberOfDivisors(m)))]

Extensions

a(13)-a(20) from Giovanni Resta, Oct 29 2019

A326310 Arithmetic numbers (A003601) that are not RMS numbers (A140480).

Original entry on oeis.org

3, 5, 6, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102
Offset: 1

Views

Author

Jaroslav Krizek, Oct 18 2019

Keywords

Comments

Numbers m such that the arithmetic mean of the divisors of m is an integer but the quadratic mean (the root mean square) of the divisors of m is not an integer.
Numbers m such that A(m) = A000203(m) / A000005(m) is an integer but Q(m) = sqrt(A001157(m) / A000005(m)) is not an integer.
Corresponding values of A(m): 2, 3, 3, 6, 7, 6, 6, 9, 10, 7, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 12, 22, 14, 13, 18, ...
Corresponding values of Q(m): sqrt(5), sqrt(13), sqrt(25/2), sqrt(61), sqrt(85), sqrt(125/2), sqrt(65), sqrt(145), sqrt(181), ...
Complement of A327055 with respect to A003601.

Links

Crossrefs

Cf.: A000005, A000203, A001157, A003601, A140480, A327055.

Programs

  • Magma
    [m: m in [1..10^6] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(Sqrt(&+[d^2: d in Divisors(m)] / NumberOfDivisors(m)))];
  • Mathematica
    Select[Range[120],IntegerQ[Mean[Divisors[#]]]&&!IntegerQ[RootMeanSquare[ Divisors[ #]]]&] (* Harvey P. Dale, Mar 04 2023 *)

A326713 Numbers m that are neither arithmetic (A003601) nor RMS numbers (A140480).

Original entry on oeis.org

2, 4, 8, 9, 10, 12, 16, 18, 24, 25, 26, 28, 32, 34, 36, 40, 48, 50, 52, 58, 63, 64, 72, 74, 75, 76, 80, 81, 82, 84, 88, 90, 98, 100, 104, 106, 108, 112, 117, 120, 121, 122, 124, 128, 130, 136, 144, 146, 148, 152, 156, 160, 162, 170, 171, 172, 175, 176, 178
Offset: 1

Views

Author

Jaroslav Krizek, Oct 18 2019

Keywords

Comments

Numbers m such that neither A(m) = A000203(m) / A000005(m) nor Q(m) = sqrt(A001157(m) / A000005(m)) is an integer.
Numbers m such that neither A(m) = A000203(m) / A000005(m) nor Q(m) = sqrt(A001157(m) / A000005(m)) is an integer.
Corresponding values of A(m): 3/2, 7/3, 15/4, 13/3, 9/2, 14/3, 31/5, 13/2, 15/2, 31/3, 21/2, 28/3, 21/2, 27/2, 91/9, 45/4, ...
Corresponding values of Q(m): sqrt(5/2), sqrt(7), sqrt(85/4), sqrt(91/3), sqrt(65/2), sqrt(35), sqrt(341/5), sqrt(455/6), ...
Sequence deviates from A049642; number 2217231104 (the first RMS number that is not arithmetic) is a term of A049642 but is not a term of this sequence.

Crossrefs

Cf. A000005, A000203, A001157, A003601, A049642, A140480, A327055.

Programs

  • Magma
    [m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(Sqrt(&+[d^2: d in Divisors(m)] / NumberOfDivisors(m)))];
  • Mathematica
    Select[Range[178], !IntegerQ @ RootMeanSquare[Divisors[#]] && !Divisible[ DivisorSigma[1, #], DivisorSigma[0, #]] &] (* Amiram Eldar, Oct 20 2019 *)
Showing 1-3 of 3 results.

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