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.

A379552 Number of pairs (d, k/d), d < k/d, such that d|k, rad(d) = rad(k/d) = rad(k), but d|k/d, for k = A376936(n), where rad = A007947.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 2, 1, 2, 1, 1, 2, 3, 4, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 4, 1, 2, 1, 3, 4, 1, 2, 6, 1, 3, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 4, 2, 2, 1, 2, 3, 1, 4, 2, 1, 1, 2, 1, 1, 3, 4
Offset: 1

Views

Author

Michael De Vlieger, Dec 25 2024

Keywords

Comments

In other words, one half the number of coreful complementary divisor pairs (d, k/d), d|k, that do not divide one another, for k in A376936, the sequence of numbers k that have at least 1 such pair.
Divisors d and k/d are both composite, further, are neither squarefree nor prime powers, hence in A126706.

Examples

			Let b(n) = A376936(n) and define property Q pertaining to (d, k/d), d|k, to be rad(d) = rad(k/d) = rad(k) but neither d | k/d nor k/d | d. Examples below show only (d, k/d) that have property Q:
a(1) = 1 since b(1) = 216 = 12*18.
a(2) = 1 since b(2) = 432 = 18*24.
a(3) = 1 since b(3) = 648 = 12*54.
a(4) = 2 since b(4) = 864 = 18*48 = 24*36.
a(14) = 3 since b(14) = 3456 = 18*192 = 36*96 = 48*72.
a(22) = 4 since b(22) = 7776 = 24*324 = 48*162 = 54*144 = 72*108, etc.

Links

Crossrefs

Cf. A007947, A126706, A376936, A379553, A379554.

Programs

  • Mathematica
    nn = 2^16;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
       rad[#1] == rad[#2],
       Mod[#1, #2] != 0,
       Mod[#2, #1] != 0] & @@ # &)], {n, Length[s]}]

A379553 Numbers k in A376936 that set records in A379552.

Original entry on oeis.org

216, 864, 3456, 7776, 31104, 124416, 279936, 497664, 972000, 1944000, 3888000, 7776000, 11664000, 15552000, 31104000, 34992000, 46656000, 62208000, 77760000, 97200000, 194400000, 291600000, 388800000, 777600000, 874800000, 1166400000, 1555200000, 3110400000, 3499200000
Offset: 1

Views

Author

Michael De Vlieger, Dec 25 2024

Keywords

Comments

Proper subset of A025487.

Examples

			Let b(n) = A376936(n) and define property Q pertaining to (d, k/d), d|k, to be rad(d) = rad(k/d) = rad(k) but neither d | k/d nor k/d | d. Table below shows prime power decomposition of a(n), n = 1..12, writing only exponents in the "exp." column:
   n       a(n)   exp.   b(n)  (d,a(n)/d) with property Q
  -----------------------------------------------------------------
   1       216    3.3      1   (12,18)
   2       864    5.3      2   (18,48), (24,36)
   3      3456    7.3      3   (18,192), (36,96), (48,72)
   4      7776    5.5      4   (24,324), (48,162), (54,144), (72,108)
   5     31104    7.5      6
   6    124416    9.5      8
   7    279936    7.7      9
   8    497664   11.5     10
   9    972000    5.5.3   12
  10   1944000    6.5.3   14
  11   3888000    7.5.3   18
  12   7776000    8.5.3   20
See expanded table in links.

Links

Crossrefs

Cf. A376936, A379552, A379554.

Programs

  • Mathematica
    r = 0; nn = 10^9;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]; nn = Length[s];
Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[k]] &@
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
    rad[#1] == rad[#2],
    Mod[#1, #2] != 0,
    Mod[#2, #1] != 0] & @@ # &)], {i, nn}] ][[-1, 1]]

A379594 Records in A379592.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21, 23, 24, 27, 28, 29, 31, 32, 34, 35, 36, 39, 41, 44, 47, 48, 49, 53, 55, 59, 62, 63, 71, 72, 74, 83, 84, 89, 95, 96, 104, 107, 111, 119, 120, 125, 127, 134, 139, 143, 149, 159, 161, 167, 179, 180
Offset: 1

Views

Author

Michael De Vlieger, Dec 30 2024

Keywords

Comments

See comments in A379593.
Compare with A036965 and A379554.

Links

Crossrefs

Cf. A036965, A379554, A379592, A379593.

Programs

  • Mathematica
    (* Load function f at A025487 *)
r = 0; s = Union@ Flatten@ f[10]; nn = Length[s];
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
  Transpose@ Reap[Monitor[
    Do[k = s[[i]];
      If[# > r, r = #; Sow[r]] &@
        Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
          _?(And[rad[#1] == rad[#2],
            Xor[Divisible[#2, #1],
                Divisible[#1, #2]]] & @@ # &)], {i, nn}], {i, nn}] ][[-1, 1]]

A379754 Records in A379752.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 12, 15, 16, 19, 21, 23, 27, 28, 37, 40, 47, 53, 57, 59, 66, 67, 69, 73, 79, 81, 85, 88, 92, 103, 117, 125, 133, 146, 147, 153, 165, 175, 185, 189, 197, 204, 227, 229, 237, 245, 269, 281, 289, 306, 311, 321, 349, 367, 393, 397, 417, 428
Offset: 1

Views

Author

Michael De Vlieger, Jan 01 2025

Keywords

Comments

See comments in A379752.

Links

Crossrefs

Cf. A379554, A379752, A379753.

Programs

  • Mathematica
    (* Load function f at A025487 *)
r = 0;
s = Select[Union@ Flatten@ f[14][[4 ;; -1]], Not@*SquareFreeQ];
nn = Length[s]; Print[nn];
Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[r]] &@
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
       Nor[Divisible[#2, rad[#1]],
           Divisible[#1, rad[#2]] ] ] & @@ # &)], {i, nn}] ][[-1, 1]]
Showing 1-4 of 4 results.

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