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 1-4 of 4 results.

A358396 Even numbers k such that sigma(k) + sigma(k+2) < 2*sigma(k+1); even terms in A053229.

Original entry on oeis.org

104, 134, 164, 314, 404, 494, 524, 554, 566, 584, 674, 692, 734, 764, 854, 944, 974, 1124, 1154, 1196, 1214, 1304, 1322, 1364, 1394, 1484, 1574, 1682, 1724, 1754, 1784, 1814, 1826, 1844, 1994, 2024, 2144, 2204, 2384, 2414, 2456, 2474, 2564, 2624, 2654, 2804, 2834, 3002
Offset: 1

Views

Author

Jianing Song, Nov 13 2022

Keywords

Comments

Even numbers k such that A053223(k) < 0.

Examples

			104 is a term since sigma(105) = 192 is greater than the average of sigma(104) = 210 and sigma(106) = 162.

Links

Crossrefs

Cf. A053229, A053223, A000203 (sigma), A358395.

Programs

  • PARI
    isA358396(n) = !(n%2) && (sigma(n) + sigma(n+2) < 2*sigma(n+1))

A053223 Second differences of sigma(n).

Original entry on oeis.org

-1, 2, -4, 7, -10, 11, -9, 7, -11, 22, -30, 24, -10, 7, -20, 34, -40, 41, -32, 14, -16, 48, -65, 40, -13, 18, -42, 68, -82, 71, -46, 21, -12, 49, -96, 75, -26, 38, -82, 102, -106, 92, -46, 0, -18, 100, -143, 103, -57, 47, -70, 110, -114, 96, -88, 50, -40, 138, -214, 140, -26, 15, -66
Offset: 1

Views

Author

Asher Auel, Jan 06 2000

Keywords

Links

Crossrefs

Cf. A000203, A053222, A053228, A053229, A053247, A053248.
Indices of 0: A099632 - 1.

Programs

  • Magma
    [DivisorSigma(1, n+2) - 2*DivisorSigma(1,n+1) + DivisorSigma(1, n): n in [1..100]]; // G. C. Greubel, Sep 03 2018
  • Mathematica
    Table[DivisorSigma[1,n+2] - 2*DivisorSigma[1,n+1] + DivisorSigma[1,n], {n,1,100}] (* G. C. Greubel, Sep 03 2018 *)
  • PARI
    a(n)=sigma(n+2)-2*sigma(n+1)+sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

Formula

G.f.: -1 - 1/x + (1 - x)^2*Sum_{k>=1} k*x^(k-2)/(1 - x^k). - Ilya Gutkovskiy, Jan 29 2017

A053228 n for which second differences of sigma(n) are positive.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Links

Crossrefs

Cf. A000203, A053223, A053229.

Programs

  • Maple
    with(numtheory): seq(`if`(sigma(i+2)-sigma(i+1)>(sigma(i+1)-sigma(i)),i,print( )), i=1..142);
  • Mathematica
    Select[Range[150], DivisorSigma[1, #+2] - DivisorSigma[1, #+1] > (DivisorSigma[1, # + 1] - DivisorSigma[1, #]) &] (* G. C. Greubel, Sep 03 2018 *)
Position[Differences[#,2]&/@Partition[DivisorSigma[1,Range[140]],3,1],?(Positive[ #]&)][[All,1]] (* _Harvey P. Dale, Jan 18 2023 *)

A053248 Second differences of sigma(n) that are negative.

Original entry on oeis.org

1, 4, 10, 9, 11, 30, 10, 20, 40, 32, 16, 65, 13, 42, 82, 46, 12, 96, 26, 82, 106, 46, 18, 143, 57, 70, 114, 88, 40, 214, 26, 66, 136, 88, 120, 244, 30, 60, 160, 171, 47, 256, 36, 150, 266, 96, 40, 286, 88, 176, 226, 124, 12, 24, 342, 170, 230, 222, 94, 34, 443, 71
Offset: 1

Views

Author

Asher Auel, Jan 11 2000

Keywords

Crossrefs

Cf. A000203, A053223, A053247.

Programs

  • Maple
    with (numtheory): f := [seq( sigma(i+2) - sigma(i+1) - (sigma(i+1) - sigma(i)), i=1..200)];
seq(`if`(f[i] < 0,f[i],print( )), i=1..200);
  • Mathematica
    Abs[Select[Differences[DivisorSigma[1,Range[200]],2],Negative]] (* Harvey P. Dale, Aug 13 2015 *)

Formula

a(n) = A053223(A053229(n)).
Showing 1-4 of 4 results.

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