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.

A362404 Numbers k such that k and k+1 are both in A362401.

Original entry on oeis.org

24, 27, 48, 79, 120, 168, 199, 288, 350, 360, 378, 391, 447, 507, 528, 775, 840, 895, 960, 1088, 1136, 1368, 1638, 1639, 1680, 1848, 1849, 2095, 2127, 2208, 2322, 2749, 2808, 3720, 3726, 3798, 3799, 3919, 4050, 4087, 4488, 4550, 4872, 5040, 5328, 5448, 5631, 6240
Offset: 1

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Author

Amiram Eldar, Apr 18 2023

Keywords

Examples

			24 is a term since 24 and 25 are both in the range of A162296: A162296(20) = 24 and A162296(25) = 25.

Links

Crossrefs

Subsequence of A362401.
A362405 is a subsequence.
Cf. A162296.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Union[Array[s, max]], 0 < # <= max &], i}, i = Position[Differences[v], 1] // Flatten; v[[i]]]; seq[10^4]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))); for(k=1, #v-1, if(v[k+1] - v[k] == 1, print1(v[k], ", ")));}

A362405 Numbers k such that k, k+1 and k+2 are all in A362401.

Original entry on oeis.org

1638, 1848, 3798, 11448, 16854, 26910, 35574, 37248, 57120, 69678, 69822, 85848, 94248, 110526, 208848, 272214, 305046, 310248, 335478, 335479, 368448, 573048, 580680, 687240, 1017126, 1154270, 1230606, 1289358, 1423248, 1467414, 1697808, 1718880, 1776750, 1777248
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Comments

Up to 10^8, k = 335478 is the only number k such that k, k+1, k+2 and k+3 are all in A362401. Are there any other such terms?

Examples

			1638 is a term since 1638, 1639 and 1640 are all in the range of A162296: A162296(1053) = 1638, A162296(576) = 1639 and A162296(1636) = 1640.

Links

Crossrefs

Subsequence of A362401 and A362404.
Cf. A162296.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Union[Array[s, max]], 0 < # <= max &], w, i, j}, i = Position[Differences[v], 1] // Flatten; w = v[[i]]; j = Position[Differences[w], 1] // Flatten; w[[j]]]; seq[10^6]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = select(x -> (x < kmax), Set(vector(kmax, k, s(k))))); for(k=1, #v-2, if(v[k+1] - v[k] == 1 && v[k+2] - v[k+1] == 1, print1(v[k], ", ")));}

A362402 Positive numbers m such that a record number of numbers k have m as the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

1, 4, 48, 72, 216, 288, 864, 1440, 1728, 2880, 3456, 4320, 5184, 5760, 8640, 12096, 17280, 25920, 34560, 48384, 51840, 69120, 103680, 120960, 155520, 181440, 207360, 241920, 311040, 362880, 483840, 622080, 725760, 967680, 1088640, 1209600, 1451520, 2177280, 2903040
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Comments

The value 0 appears in the range of A162296 for all squarefree numbers (A005117) and therefore it is excluded from this sequence.
The corresponding record values are in A362403.
Except for 1, a subsequence of A362401.

Links

Crossrefs

Cf. A162296, A362401, A362403.
Similar sequences: A097942, A100827, A145899, A238895.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {1}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[1]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(1, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", "))); }

A362403 Number of times that the number A362402(n) occurs as a sum of divisors that have a square factor (A162296).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 10, 13, 15, 16, 20, 22, 23, 28, 34, 46, 53, 60, 62, 78, 81, 113, 115, 122, 132, 154, 184, 185, 222, 248, 254, 343, 346, 350, 354, 497, 569, 701, 711, 860, 941, 1088, 1221, 1222, 1235, 1263, 1306, 1572, 1721, 1737, 1948, 2191, 2315, 2418, 2877
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Links

Crossrefs

Cf. A162296, A362401, A362402.
Similar sequences: A131934, A101373, A206027, A238896.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {0}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[2]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(0, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(v[k], ", "))); }
Showing 1-4 of 4 results.

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