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.

A335254 Numbers k such that the abundance (A033880) of k is equal to the deficiency (A033879) of k+1.

Original entry on oeis.org

672, 523776, 19327369215
Offset: 1

Views

Author

Amiram Eldar, May 28 2020

Keywords

Comments

Equivalently, k and k+1 have the same absolute value of abundance (or deficiency) with opposite signs.
Equivalently, s(k) + s(k+1) = k + (k+1), where s(k) is the sum of proper divisors of k (A001065).
If k is a 3-perfect number (A005820) and k+1 is a prime, then k is in the sequence. Of the 6 known 3-perfect numbers only 672 and 523776 have this property.
a(4) > 10^11, if it exists.
a(4) > 10^13, if it exists. - Giovanni Resta, May 30 2020

Examples

			672 is a term since A033880(672) = sigma(672) - 2*672 = 2016 - 1344 = 672, and A033879(673) = 2*673 - sigma(673) = 1346 - 674 = 672.

Crossrefs

Cf. A000203, A001065, A005820, A033879, A033880, A330901, A335253.

Programs

  • Mathematica
    ab[n_] := DivisorSigma[1, n] - 2*n; Select[Range[6 * 10^5], ab[#] == -ab[# + 1] &]

A335252 Numbers k such that k and k+2 have the same unitary abundance (A129468).

Original entry on oeis.org

12, 63, 117, 323, 442, 1073, 1323, 1517, 3869, 5427, 6497, 12317, 18419, 35657, 69647, 79919, 126869, 133787, 151979, 154007, 163332, 181427, 184619, 333797, 404471, 439097, 485237, 581129, 621497, 825497, 1410119, 2696807, 3077909, 3751619, 5145341, 6220607
Offset: 1

Views

Author

Amiram Eldar, May 28 2020

Keywords

Comments

Are 12, 442 and 163332 the only even terms?
Are there any unitary abundant numbers (A034683) in this sequence?
No further even terms up to 10^13. - Giovanni Resta, May 30 2020

Examples

			12 is a term since 12 and 14 have the same unitary abundance: A129468(12) = usigma(12) - 2*12 = 20 - 24 = -4, and A129468(14) = usigma(14) - 2*14 = 24 - 28 = -4.

Links

Crossrefs

The unitary version of A330901.
Cf. A034448, A034683, A129468, A129487, A335251.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); udef[n_] := 2*n - usigma[n]; Select[Range[10^5], udef[#] == udef[# + 2] &]

A335253 Numbers k such that the abundance (A033880) of k is equal to the deficiency (A033879) of k+2.

Original entry on oeis.org

12, 76, 488, 556, 1100, 1430, 2408, 8896, 538208, 13685780, 962402768
Offset: 1

Views

Author

Amiram Eldar, May 28 2020

Keywords

Comments

Equivalently, k and k+2 have the same absolute value of abundance (or deficiency) with opposite signs.
Equivalently, s(k) + s(k+2) = k + (k+2), where s(k) is the sum of proper divisors of k (A001065).
a(12) > 10^11, if it exists.
a(12) > 10^13, if it exists. - Giovanni Resta, May 30 2020

Examples

			12 is a term since A033880(12) = sigma(12) - 2*12 = 28 - 24 = 4, and A033879(14) = 2*14 - sigma(14) = 28 - 24 = 4.

Crossrefs

Cf. A000203, A001065, A033879, A033880, A330901, A335254.

Programs

  • Mathematica
    ab[n_] := DivisorSigma[1, n] - 2*n; Select[Range[10^5], ab[#] == -ab[# + 2] &]
  • PARI
    isok(k) = sigma(k) + sigma(k+2) == 4*k+4; \\ Michel Marcus, May 29 2020
Showing 1-3 of 3 results.

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

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