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.

A328952 Arithmetic numbers (A003601) that are not antiharmonic (A020487).

Original entry on oeis.org

3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 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, Nov 17 2019

Keywords

Comments

Numbers m such that the arithmetic mean of the divisors of m is an integer but the antiharmonic mean of the divisors of m is not an integer.
Numbers m such that A(m) = A000203(m) / A000005(m) is an integer but B(m) = A001157(m) / A000203(m) is not an integer.
Corresponding values of A(m): 2, 3, 3, 4, 6, 7, 6, 6, 9, 10, 8, 9, 12, 10, 15, 9, 16, 12, 12, 19, 15, 14, 21, 12, 22, ...
Corresponding values of B(m): 5/2, 13/3, 25/6, 25/4, 61/6, 85/7, 125/12, 65/6, 145/9, 181/10, 125/8, ...

Crossrefs

Complement of A277553 with respect to A003601.
Cf. A000005, A000203, A001157, A328953, A328954.

Programs

  • Magma
    [m: m in [1..10^5] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m))]
  • Mathematica
    Select[Range[100], Divisible[(sigma = DivisorSigma[1, #]), DivisorSigma[0, #]] && !Divisible[DivisorSigma[2, #], sigma] &]  (* Amiram Eldar, Nov 17 2019 *)
  • PARI
    isok(m) = !(sigma(m) % numdiv(m)) && (sigma(m,2) % sigma(m)); \\ Michel Marcus, Nov 18 2019

A328953 Antiharmonic numbers (A020487) that are not arithmetic (A003601).

Original entry on oeis.org

4, 9, 16, 25, 36, 50, 64, 81, 100, 117, 121, 144, 180, 196, 200, 225, 242, 256, 289, 324, 325, 400, 441, 450, 468, 484, 529, 576, 578, 625, 650, 676, 729, 784, 800, 841, 900, 968, 1024, 1058, 1089, 1156, 1225, 1280, 1296, 1300, 1444, 1476, 1521, 1600, 1620
Offset: 1

Views

Author

Jaroslav Krizek, Nov 17 2019

Keywords

Comments

Numbers m such that the antiharmonic mean 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 B(m) = A001157(m) / A000203(m) is an integer but A(m) = A000203(m) / A000005(m) is not an integer.
Corresponding values of B(m): 3, 7, 11, 21, 21, 35, 43, 61, 63, 85, 111, 77, 91, 129, 119, 147, 185, 171, 273, 183, ...
Corresponding values of A(m): 7/3, 13/3, 31/5, 31/3, 91/9, 31/2, 127/7, 121/5, 217/9, 91/3, 133/3, ...

Crossrefs

Complement of A277553 with respect to A020487.
Cf. A000005, A000203, A001157, A003601, A328952, A328954.

Programs

  • Magma
    [m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m))]
  • Mathematica
    Select[Range[1620], !Divisible[(sigma = DivisorSigma[1, #]), DivisorSigma[0, #]] && Divisible[DivisorSigma[2, #], sigma] &] (* Amiram Eldar, Nov 17 2019 *)

A328954 Numbers m that are neither arithmetic (A003601) nor antiharmonic (A020487).

Original entry on oeis.org

2, 8, 10, 12, 18, 24, 26, 28, 32, 34, 40, 48, 52, 58, 63, 72, 74, 75, 76, 80, 82, 84, 88, 90, 98, 104, 106, 108, 112, 120, 122, 124, 128, 130, 136, 146, 148, 152, 156, 160, 162, 170, 171, 172, 175, 176, 178, 192, 194, 202, 208, 216, 218, 226, 228, 232, 234
Offset: 1

Views

Author

Jaroslav Krizek, Dec 03 2019

Keywords

Comments

Numbers m such that neither the arithmetic mean of the divisors of m nor the antiharmonic mean of the divisors of m is an integer.
Numbers m such that neither A(m) = A000203(m)/A000005(m) nor B(m) = A001157(m)/A000203(m) is an integer.
Corresponding values of A(m): 3/2, 15/4, 9/2, 14/3, 13/2, 15/2, 21/2, 28/3, 21/2, 27/2, 45/4, 62/5, ...
Corresponding values of B(m): 5/3, 17/3, 65/9, 15/2, 35/3, 85/6, 425/21, 75/4, 65/3, 725/27, 221/9, ...

Crossrefs

Cf. A000005, A000203, A001157, A003601, A277553, A328952, A328953.

Programs

  • Magma
    [m: m in [1..10^5] | not IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and not IsIntegral(&+[d^2: d in Divisors(m)] / SumOfDivisors(m))]
  • Mathematica
    Select[Range[235], !Divisible[DivisorSigma[2, #], (s = DivisorSigma[1, #])] && !Divisible[s, DivisorSigma[0, #]] &] (* Amiram Eldar, Dec 06 2019 *)
Showing 1-3 of 3 results.

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

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