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.

A345051 Numbers k such that A345048(k) is equal to A345049(k).

Original entry on oeis.org

2, 6, 9, 15, 28, 496, 625, 1225, 3993, 8128, 117649, 218491, 857375, 3788435, 4259571, 33550336, 69302975, 136410197, 200533921, 313742585, 603439225, 1516358753, 2563893625, 3326174929, 5655792025, 8589869056, 10214476341
Offset: 1

Views

Author

Antti Karttunen, Jun 06 2021

Keywords

Comments

Numbers k for which A342001(n) * A051709(n) = A173557(n) * A345001(n).
Conjecture: Sequence is a disjoint union of A000396 and A166374, i.e., there are no terms of any other kind.

Links

Crossrefs

Cf. A051709, A173557, A342001, A345001, A345048, A345049.
Positions of zeros in A345050.
Cf. A000396, A166374 (subsequences).
Cf. also A345003.

Programs

Extensions

a(21)-a(27) from Amiram Eldar, Dec 08 2023

A345050 a(n) = A345048(n) - A345049(n).

Original entry on oeis.org

1, 0, 2, -1, 12, 0, 30, -2, 0, -6, 90, 24, 132, -12, 0, -3, 240, 15, 306, 16, 20, -24, 462, 108, 38, -30, -14, 0, 756, 276, 870, -4, 108, -42, 264, 152, 1260, -48, 176, 130, 1560, 348, 1722, -56, -60, -60, 2070, 336, 164, -35, 360, -96, 2652, 84, 912, 136, 476, -78, 3306, 1824, 3540, -84, -110, -5, 1380, 492, 4290
Offset: 1

Views

Author

Antti Karttunen, Jun 06 2021

Keywords

Crossrefs

Cf. A345048, A345049, A345051 (positions of zeros).

Programs

Formula

a(n) = A345048(n) - A345049(n).

A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

Original entry on oeis.org

-1, 0, -1, 3, -3, 5, -5, 11, 1, 5, -9, 20, -11, 5, 2, 31, -15, 24, -17, 26, 0, 5, -21, 56, -9, 5, 13, 32, -27, 43, -29, 79, -4, 5, -10, 79, -35, 5, -6, 78, -39, 53, -41, 44, 27, 5, -45, 140, -27, 38, -10, 50, -51, 93, -22, 100, -12, 5, -57, 140, -59, 5, 29, 191, -28, 73, -65, 62, -16, 63, -69, 207, -71, 5, 29, 68
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2021

Keywords

Comments

Coincides with A003415 only on perfect numbers (A000396).

Crossrefs

Cf. A000203, A000396, A001065, A003415, A033879, A168036, A344999, A345002, A345003, A345004, A345005, A345049.
Cf. also A051709, A344753 (analogous sequences).
Cf. A013661, A136141.

Programs

  • Mathematica
    A003415[n_] := If[n < 2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
a[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
Array[a, 80] (* Jean-François Alcover, Jun 12 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));

Formula

a(n) = A003415(n) - A033879(n) = A000203(n) + A003415(n) - 2*n.
a(n) = A001065(n) + A168036(n).
a(n) = A344999(n) / A048250(n) = A345049(n) / A173557(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 + A136141 - 2 = 0.418090735898... . - Amiram Eldar, Dec 08 2023

A345048 a(n) = A342001(n) * A051709(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 10, 0, 9, 2, 14, 0, 64, 0, 18, 16, 28, 0, 63, 0, 120, 20, 26, 0, 220, 2, 30, 12, 192, 0, 620, 0, 75, 28, 38, 24, 310, 0, 42, 32, 442, 0, 984, 0, 384, 156, 50, 0, 616, 2, 117, 40, 504, 0, 270, 32, 736, 44, 62, 0, 2944, 0, 66, 238, 186, 36, 1952, 0, 792, 52, 1652, 0, 975, 0, 78, 154, 960, 36, 2556, 0, 1276, 52
Offset: 1

Views

Author

Antti Karttunen, Jun 06 2021

Keywords

Links

Crossrefs

Cf. A345049, A345050, A345051.

Programs

Formula

a(n) = A342001(n) * A051709(n).
a(n) = A345049(n) + A345050(n).
Showing 1-4 of 4 results.

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