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 1-7 of 7 results.

A383269 a(n) is the smallest nonnegative solution to sigma(A383268(n) - x) + sigma(A383268(n) + x) = 4*A383268(n).

Original entry on oeis.org

0, 1, 5, 11, 0, 7, 17, 28, 26, 37, 23, 14, 7, 13, 17, 49, 11, 22, 11, 5, 1, 58, 70, 13, 20, 37, 19, 11, 17, 31, 41, 67, 6, 16, 13, 73, 49, 11, 55, 91, 19, 73, 119, 5, 11, 77, 53, 43, 103, 86, 7, 114, 173, 88, 71, 59, 124, 95, 139, 7, 128, 31, 92, 143, 83, 227, 163
Offset: 1

Views

Author

Felix Huber, Apr 24 2025

Keywords

Examples

			a(2) = 1 because sigma(A383268(2) - 1) + sigma(A383268(2) + 1) =  sigma(13 - 1) + sigma(13 + 1) = sigma(12) + sigma(14) = 28 + 24 = 52 = 4*13 = 4*A383268(2).

Links

Crossrefs

Cf. A000203, A000396, A186103, A383268.

Programs

  • Maple
    with(NumberTheory):
A383269:=proc(N) # To get the first N terms.
    local k,x,X;
    X:=[];
    for k while nops(X)A383269(67);
  • PARI
    isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 4*k, return(x))); return(-1);
lista(nn) = for (n=1, nn, my(m=isok(n)); if (m != -1, print1(m, ", "))); \\ Michel Marcus, Apr 26 2025

Formula

a(n) = 0 iff A383268(n) is a perfect number (A000396) and vice versa.

A383758 Least integer k for which sigma(k - x) + sigma(k + x) = n*k has at least one solution.

Original entry on oeis.org

1, 2, 6, 24, 93, 1952, 14412, 361881, 61824672
Offset: 2

Views

Author

Jean-Marc Rebert, May 09 2025

Keywords

Comments

The corresponding x are : 0, 0, 0, 0, 87, 1828, 13308, 358839, ...
a(10) <= 61824672 via sigma(61824672 - 60697728) + sigma(61824672 + 60697728) = 10*61824672. - Michel Marcus, May 20 2025
a(11) <= 43293761280 via sigma(43293761280 - 40511560320) + sigma(43293761280 + 40511560320) == 11*43293761280. - Michel Marcus, May 25 2025
Note that for n=2,3,4,5,8,and 9, we have k+x = A383920(n). - Michel Marcus, Jun 09 2025
From David A. Corneth, Jun 13 2025: (Start)
a(10) = 61824672. We must have sigma(k-x) >= 5*(k-x) or sigma(k+x) >= 5 * (k+x).
The numbers <= 2*61824672 that have this property are 122522400. It has been checked that if k + x = 122522400 then k must be 61824672 to get the smallest such k. (End)

Examples

			a(4) = 6 because the equation sigma(6-x) + sigma(6+x) = 4*6 has the solution x = 0 and no smaller number possesses this property. See A000396, A383268, and A383269.
a(5) = 24 because the equation sigma(24-x) + sigma(24+x) = 5*24 has the solution x = 0. This is verified as follows: sigma(24-0) + sigma(24+0) = sigma(24) + sigma(24) = 60 + 60 = 120 = 5*24. Moreover, no smaller number possesses this property. See A141643.
a(6) = 93 because the equation sigma(93 - x) + sigma(93 + x) = 6 * 93 has the solution x = 87: sigma(93 - 87) + sigma(93 + 87) = sigma(6) + sigma(180) = 12 + 546 = 6*93. Moreover, no smaller number possesses this property.

Crossrefs

Cf. A000203, A000396, A141643, A317681, A383268, A383269.

Programs

  • PARI
    isok(k,n) = forstep(x=k-1, 0, -1, if (sigma(k - x) + sigma(k + x) == n*k, return(1)));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, May 10 2025

Formula

a(n) <= A317681(n).

Extensions

a(10) from Michel Marcus and David A. Corneth, Jun 13 2025

A384838 Numbers k for which sigma(k - x) + sigma(k + x) = 5*k has at least one nonnegative solution.

Original entry on oeis.org

24, 53, 56, 63, 66, 74, 75, 79, 82, 84, 95, 112, 168, 192, 216, 227, 245, 252, 255, 274, 280, 282, 288, 308, 312, 347, 348, 351, 360, 365, 392, 395, 408, 420, 431, 432, 434, 458, 465, 466, 471, 476, 496, 528, 532, 560, 576, 579, 588, 624, 628, 644, 670, 694, 716, 720, 784
Offset: 1

Views

Author

Michel Marcus, Jun 10 2025

Keywords

Links

Crossrefs

Cf. A000203 (sigma), A141643 (a subsequence).
Cf. A383268, A384839, A384840, A384841.
Cf. A383758.

Programs

  • PARI
    isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 5*k, return(1)));

A384839 Numbers k for which sigma(k - x) + sigma(k + x) = 6*k has at least one nonnegative solution.

Original entry on oeis.org

93, 120, 204, 211, 231, 239, 254, 269, 274, 280, 315, 336, 343, 360, 366, 372, 375, 378, 395, 396, 402, 420, 466, 475, 496, 592, 604, 672, 708, 724, 726, 774, 821, 822, 827, 836, 840, 844, 845, 862, 870, 880, 898, 919, 926, 952, 964, 976, 982, 996, 997, 1023, 1077, 1080
Offset: 1

Views

Author

Michel Marcus, Jun 10 2025

Keywords

Links

Crossrefs

Cf. A000203 (sigma), A005820 (a subsequence).
Cf. A383268, A384838, A384840, A384841.
Cf. A383758.

Programs

  • Mathematica
    s={};Do[Do[If[DivisorSigma[1,k-x]+DivisorSigma[1,k+x]==6*k,AppendTo[s,k];Break[]],{x,0,k-1}],{k,1,1080}];s (* James C. McMahon, Jun 20 2025 *)
  • PARI
    isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 6*k, return(1)));

A384840 Numbers k for which sigma(k - x) + sigma(k + x) = 7*k has at least one nonnegative solution.

Original entry on oeis.org

1952, 1992, 2016, 2160, 2520, 2640, 3314, 3744, 3801, 4320, 4484, 4500, 4566, 4620, 4680, 4994, 5016, 5948, 6072, 6096, 6194, 6384, 6492, 6552, 6654, 6870, 7056, 7200, 7224, 7382, 7435, 7440, 7470, 7794, 7812, 7920, 7956, 8376, 8480, 8604, 8616, 8702, 8892, 9284, 9360, 9408
Offset: 1

Views

Author

Michel Marcus, Jun 10 2025

Keywords

Links

Crossrefs

Cf. A000203 (sigma), A055153 (a subsequence).
Cf. A383268, A384838, A384839, A384841.
Cf. A383758.

Programs

  • PARI
    isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 7*k, return(1)));

A384841 Numbers k for which sigma(k - x) + sigma(k + x) = 8*k has at least one nonnegative solution.

Original entry on oeis.org

14412, 17640, 25581, 25623, 25659, 26208, 30240, 31110, 31380, 31500, 32340, 32736, 32760, 34958, 35112, 44211, 44343, 45048, 45324, 45444, 46578, 48090, 49368, 51674, 52045, 52290, 53103, 53127, 53460, 54000, 54180, 59400, 59940, 60228, 60903, 60914, 60987, 61920, 62340, 62370
Offset: 1

Views

Author

Michel Marcus, Jun 10 2025

Keywords

Links

Crossrefs

Cf. A000203 (sigma), A027687 (a subsequence).
Cf. A383268, A384838, A384839, A384840.
Cf. A383758.

Programs

  • PARI
    isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 8*k, return(1)));

A385075 Numbers k for which sigma(k - x) + sigma(k + x) = 9*k has at least one nonnegative solution.

Original entry on oeis.org

361881, 792960, 835072, 837312, 846720, 917280, 944608, 946176, 1509236, 1517264, 1630880, 1635600, 1636614, 1697560, 1834560, 1914092, 1926336, 1927692, 1941264, 2387120, 2450112, 2474316, 2494464, 2546656, 2573088, 2624832, 2685394, 2705680, 2840468, 2913120, 2941009
Offset: 1

Views

Author

Michel Marcus, Jun 16 2025

Keywords

Comments

At least one of sigma(k - x) >= 4.5*(k - x) or sigma(k+x) >= 4.5*(k + x) is true. - David A. Corneth, Jun 16 2025

Examples

			361881 is in the sequence via k = 361881 and x = 358839 so k-x = 3042, k+x = 720720 and sigma(k - x) + sigma(k + x) = 7137 + 3249792 = 3256929 = 9*361881 = 9*k. - _David A. Corneth_, Jun 16 2025

Links

Crossrefs

Cf. A000203, A141645 (a subsequence).
Cf. A383268, A384838, A384839, A384840, A384841.
Cf. A383758.

Programs

  • PARI
    isok(k) = forstep(x=k-1, 0, -1, if (sigma(k - x) + sigma(k + x) == 9*k, return(1)));

Extensions

More terms from David A. Corneth, Jun 16 2025
Showing 1-7 of 7 results.

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