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.

A357608 Numbers k such that k and k+1 are both in A357605.

Original entry on oeis.org

76544, 104895, 126224, 165375, 170624, 174824, 201824, 245024, 257984, 271215, 273104, 316575, 338624, 387855, 447615, 469664, 477224, 540224, 618975, 633555, 641024, 659295, 705375, 752895, 770175, 842624, 843975, 862784, 870975, 893024, 913275, 957824, 1047375
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

Numbers k such that A162296(k) > 2*k and A162296(k+1) > 2*(k+1).

Examples

			76544 is a term since 76544 and 76545 are both in A357605: A162296(76544) = 170688 > 2*76544 and A162296(76545) = 157248 > 2*76545.

Links

Crossrefs

Cf. A162296.
Subsequence of A013929, A096399 and A357605.
Similar sequences: A096399, A283418, A318167, A327635, A327942, A331412, A333951.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[2, 10^6], q[#] && q[#+1] &]

A357606 Primitive terms of A357605: numbers in A357605 with no proper divisor in A357605.

Original entry on oeis.org

36, 48, 80, 120, 162, 168, 200, 224, 264, 270, 280, 300, 312, 352, 378, 392, 408, 416, 450, 456, 500, 552, 588, 594, 630, 696, 700, 702, 744, 750, 882, 888, 918, 968, 980, 984, 1026, 1032, 1050, 1088, 1100, 1128, 1216, 1232, 1242, 1272, 1300, 1372, 1416, 1452
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

Numbers k such that A162296(k) > 2*k but for all the aliquot divisors d of k (i.e., d | k, d < k), A162296(d) <= 2*d.
If k is a term then all the positive multiples of k are terms of A357605.
The least odd term is a(144) = 4725.

Examples

			36 is a term since A162296(36) = 79 > 2*36, but for all the divisors d of 36, 1, 2, 3, 4, 6, 9, 12 and 18, A162296(d) <= 2*d. E.g., A162296(18) = 28 < 2*18.

Links

Crossrefs

Cf. A162296.
Subsequence of A005101, A013929 and A357605.
Similar sequences: A091191, A302574.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; q[1] = False; primQ[n_] := q[n] && AllTrue[Most @ Divisors[n], ! q[#] &]; Select[Range[1500], primQ]

A357609 Numbers k such that k, k+1, and k+2 are all in A357605.

Original entry on oeis.org

10667829248, 14322877568, 25929352448, 26967189248, 31315096448, 32186016224, 35337613310, 36312573374, 41326711424, 53162460350, 69405075584, 71840196350, 72806666750, 73217757248, 83103523424, 106184935934, 109302242048, 111640866974, 115294917374, 116768901248
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

Numbers k such that A162296(k) > 2*k, A162296(k+1) > 2*(k+1), and A162296(k+2) > 2*(k+2).

Examples

			10667829248 is a term since 10667829248, 10667829249 and 10667829250 are all in A357605: A162296(10667829248) = 21342038784 > 2*10667829248, A162296(10667829249) = 21798236160 > 2*10667829249 and A162296(10667829250) = 21810824640 > 2*10667829250.

Links

Crossrefs

Cf. A162296.
Subsequence of A013929, A096536, A357605 and A357608.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; v = Cases[Import["https://oeis.org/A096536/b096536.txt", "Table"], {, }][[;; , 2]]; s = {}; Do[n = v[[k]]; If[q[n] && q[n+1] && q[n+2], AppendTo[s, n]], {k, 1, Length[v]}]; s

A357607 Odd numbers k such that A162296(k) > 2*k.

Original entry on oeis.org

4725, 6615, 7875, 8505, 11025, 14175, 15435, 17325, 19845, 20475, 22275, 23625, 24255, 25515, 26775, 28665, 29925, 31185, 33075, 36225, 36855, 37125, 37485, 38115, 39375, 40425, 41895, 42525, 46305, 47775, 48195, 50715, 51975, 53235, 53865, 55125, 57915, 59535
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

The least term that is not divisible by 3 is a(89047132) = 134785275625.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 4, 60, 640, 6650, 66044, 660230, 6604594, 66073470, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000660... .

Examples

			4725 is a term since it is odd, and A162296(4725) = 9728 > 2*4725.

Links

Crossrefs

Cf. A162296.
Subsequence of A005231, A013929 and A357605.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275, A348605.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[3, 60000, 2], q]
Showing 1-4 of 4 results.

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

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