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.

Previous Showing 11-15 of 15 results.

A351853 Numbers that are divisible by the number of their divisors over the Eisenstein integers.

Original entry on oeis.org

1, 2, 3, 6, 8, 24, 40, 80, 81, 88, 120, 128, 136, 162, 180, 184, 225, 232, 240, 264, 324, 328, 360, 376, 384, 408, 424, 448, 450, 472, 552, 560, 568, 625, 640, 648, 664, 696, 712, 756, 808, 856, 880, 896, 900, 904, 984, 1040, 1048, 1096, 1128, 1192, 1250, 1272
Offset: 1

Views

Author

Amiram Eldar, Feb 22 2022

Keywords

Comments

Numbers k such that A319442(k) | k.
All the odd terms are squares or numbers of the form 3 times a square.

Examples

			6 is a term since it is divisible by A319442(6) = 6.

Links

Crossrefs

Cf. A033950, A048166, A319442, A351851.

Programs

  • Mathematica
    f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[n, eisNumDiv[n]]; Select[Range[1000], q]

A323392 Positive integers that have a record number of divisors in Eisenstein integers.

Original entry on oeis.org

1, 2, 3, 6, 12, 18, 21, 36, 42, 84, 126, 168, 252, 420, 504, 546, 1008, 1092, 1638, 2184, 3276, 5460, 6552, 7644, 9828, 10374, 13104, 15288, 16380, 20748, 31122, 38220, 41496, 62244, 103740, 124488, 145236, 186732, 207480, 248976, 290472, 311220, 435708, 622440, 726180, 871416
Offset: 1

Views

Author

Jianing Song, Jan 13 2019

Keywords

Comments

Indices of records in A319442.
Analog of A002182 and A279254, which list the positive integers that have a record number of divisors in rational integers and Gaussian integers respectively.
It seems that 21 is the largest odd term.

Examples

			252 has 60 divisors up to association in Eisenstein integers, more than any previous positive integers, so 252 is a term.

Links

Crossrefs

Cf. A002182, A279254, A319442.
For the number of divisors of a(n) see A323393.

Programs

  • Maple
    vmax:= 0: recinds:= NULL:
for n from 1 to 100000 do
    v := A319442(n);
    if v > vmax then vmax:= v; recinds:= recinds, n fi
od:
recinds; # Peter Luschny, Jan 19 2019
  • Mathematica
    f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; emax = 0; Do[eis = eisNumDiv[n]; If[eis > emax, emax = eis; AppendTo[seq, n]], {n, 1, 10^6}]; seq (* Amiram Eldar, Mar 02 2020 *)
  • PARI
    my(r=0, t); for(n=1, 10^6, t=A319442(n); if(t>r, r=t; print1(n, ", ")));

A323393 a(n) is the number of divisors of A323392(n) in Eisenstein integers.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 12, 15, 24, 36, 40, 48, 60, 72, 80, 96, 100, 144, 160, 192, 240, 288, 320, 324, 336, 384, 400, 432, 480, 576, 640, 648, 768, 960, 1152, 1280, 1296, 1344, 1536, 1600, 1728, 1920, 2160, 2560, 2592, 2880, 3200, 3456, 3600, 3840, 4320, 4608, 5120, 5760, 6144, 6400, 7200, 7680
Offset: 1

Views

Author

Jianing Song, Jan 13 2019

Keywords

Comments

Records in A319442.
Analog of A002183 and A302249, which list the records of number of divisors in rational integers and Gaussian integers respectively.
It seems that 15 is the largest odd term.

Examples

			252 has 60 divisors up to association in Eisenstein integers, more than any previous positive integers, so 60 is a term.

Links

Crossrefs

Cf. A002183, A302249, A319442.
For the numbers whose number of divisors set new records see A323392.

Programs

  • Mathematica
    f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; emax = 0; Do[eis = eisNumDiv[n]; If[eis > emax, emax = eis; AppendTo[seq, eis]], {n, 1, 10^6}]; seq (* Amiram Eldar, Mar 02 2020 *)
  • PARI
    my(r=0, t); for(n=1, 10^6, t=A319442(n); if(t>r, r=t; print1(r, ", ")));

Formula

a(n) = A319442(A323392(n)).

A351854 Numbers k such that k and k+1 are both divisible by the number of their divisors over the Eisenstein integers.

Original entry on oeis.org

1, 2, 80, 3968, 50624, 497024, 505520, 3207680, 6890624, 9150624, 12383360, 12852224, 13549760, 19210688, 20657024, 25250624, 41796224, 41873840, 47900240, 48650624, 79121024, 81450624, 86099840, 132503120, 140920640, 149450624, 174636224, 186732224, 214769024
Offset: 1

Views

Author

Amiram Eldar, Feb 22 2022

Keywords

Comments

Numbers k such that A319442(k) | k and A319442(k+1) | k+1.
Except for 1 and 2, all the terms are even numbers of the form k^2 - 1 (A033996).

Examples

			2 is a term since 2 is divisible by A319442(2) = 2 and 3 is divisible by A319442(3) = 3.
80 is a term since 80 is divisible by A319442(80) = 10 and 81 is divisible by A319442(81) = 9.

Links

Crossrefs

Subsequence of A351853.
Cf. A033996, A114617, A319442, A351852.

Programs

  • Mathematica
    f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[n, eisNumDiv[n]]; Join[{1, 2}, Select[Range[3, 15000, 2]^2 - 1, q[#] && q[# + 1] &]]

A323384 Smallest number with exactly n divisors in Eisenstein integers.

Original entry on oeis.org

1, 2, 3, 7, 9, 6, 27, 14, 12, 18, 243, 21, 729, 54, 36, 56, 6561, 60, 19683, 63, 108, 486, 177147, 42, 144, 1458, 147, 189, 4782969, 180, 14348907, 182, 972, 13122, 432, 84, 387420489, 39366, 2916, 126, 3486784401, 540, 10460353203, 1701, 441, 354294, 94143178827, 168, 1728, 720
Offset: 1

Views

Author

Jianing Song, Jan 12 2019

Keywords

Comments

a(n) is the smallest k such that A319442(k) = n.
Analog of A005179 and A302252 over the ring of Eisenstein integers. The divisors are counted up to association.

Examples

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
The divisors of 14 in Eisenstein integers are 1, 2, 2 + w, 2 + w', 7, 4 + 2*w, 4 + 2*w', 14 and there associations, and 14 is the smallest number having exactly 8 divisors in Eisenstein integers, so a(8) = 14.
The divisors of 21 in Eisenstein integers are 1, 2*w - 1, 3, 2 + w, 2 + w', 5 - w, 5 - w', 6 + 3*w, 6 + 3*w', 7, 14*w - 7, 21 and there associations, and 21 is the smallest number having exactly 12 divisors in Eisenstein integers, so a(12) = 21.

Crossrefs

Cf. A005179, A302252, A319442 (number of divisors of n in Eisenstein integers).

Programs

  • PARI
    a(n) = if(isprime(n)&&n>2, 3^((n-1)/2), my(k=1); while(A319442(k)!=n, k++); k)

Formula

For primes p > 2, a(p) = 3^((p-1)/2).
Previous Showing 11-15 of 15 results.

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