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 1-5 of 5 results.

A379753 Numbers that set records in A379752.

Original entry on oeis.org

60, 120, 240, 480, 840, 1260, 1680, 2520, 3360, 5040, 7560, 10080, 15120, 20160, 27720, 36960, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 443520, 498960, 554400, 665280, 720720, 997920, 1081080, 1330560, 1441440, 2162160, 2882880, 3603600, 4324320, 5765760
Offset: 1

Views

Author

Michael De Vlieger, Jan 01 2025

Keywords

Comments

Proper subset of the intersection of A025487 and A375055.
Conjecture: subset of A332785 = A126706 \ A286708.
This sequence seems to be rich in highly composite numbers, the prime shape of a(n) resembles that of highly composite numbers, with long tails of large prime factors with multiplicity 1.
Terms not in A002182 are not all of the form 2^5 * prime(i..j), 1 < i < j, for example, a(24) = 443520 = 2^7 * 3^2 * 5 * 7 * 11.

Examples

			Let b(n) = A379752(n).
Table showing exponents of prime power factors of a(n) for n = 1..12.
Example: a(6) = 1260 = 2^2 * 3^2 * 5 * 7, hence we write "2.2.1.1".
   n      a(n)       Exp.    b(a(n))
  ----------------------------------
   1       60 **   2.1.1        1   6*10
   2      120 **   3.1.1        2   6*20 = 10*12
   3      240 *    4.1.1        3   6*40 = 10*24 = 12*20
   4      480      5.1.1        4   6*80 = 10*48 = 12*40 = 20*24
   5      840 *    3.1.1.1      6   6*140 = 10*84 = 12*70 = 14*60 = 20*42 = 28*30
   6     1260 *    2.2.1.1      7
   7     1680 *    4.1.1.1      9
   8     2520 **   3.2.1.1     11
   9     3360      5.1.1.1     12
  10     5040 **   4.2.1.1     15
  11     7560 *    3.3.1.1     16
  12    10080 *    5.2.1.1     19
*  = a(n) is highly composite (in A002182),
** = a(n) is superior highly composite (in both A002182 and A002201).

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..226
  • Michael De Vlieger, Prime Power Decomposition of a(n) for n = 1..226, expanding on the table in the example.
  • Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.

Crossrefs

Cf. A002182, A002201, A025487, A375055, A379752, A379754.

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[k]] &@
  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]]

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]]

A376281 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d}, where k is in A379336.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 3, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1
Offset: 1

Views

Author

Michael De Vlieger, Jan 08 2025

Keywords

Comments

Number of ways to write k = A379336(n) as a product of numbers i and j that are neither coprime nor does one number divide the other. Both i and j are necessarily composite.
Both i and j = k/i appear in row k of A133995.

Examples

			Let s(n) = A379336(n).
a(1) = 1 since s(1) = 24 = 4*6.
a(2) = 1 since s(2) = 40 = 4*10.
a(3) = 1 since s(3) = 48 = 6*8.
a(12) = 2 since s(12) = 96 = 6*16 = 8*12.
a(16) = 3 since s(16) = 120 = 4*30 = 6*20 = 10*12.
a(44) = 4 since s(44) = 240 = 6*40 = 8*30 = 10*24 = 12*20.
a(75) = 5 since s(75) = 360 = 4*90 = 10*36 = 12*30 = 15*24 = 18*20.
a(105) = 6 since s(105) = 480 = 6*80 = 8*60 = 10*48 = 12*40 = 16*30 = 20*24, etc.

Links

Crossrefs

Cf. A045763, A133995, A379336, A379552, A379752, A379772.

Programs

  • Mathematica
    nn = 500; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
s = Complement[
  Select[Range[nn],
    And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
  Union[Reap[
    While[p <= mm, q = NextPrime[p];
      While[p*q <= mm, If[p != q, Sow[p*q]]; q = NextPrime[q]];
        p = NextPrime[p]] ][[-1, 1]] ]^2 ];
Table[k = s[[n]];
  1/2*DivisorSum[k, 1 &, ! MemberQ[{1, #1, #2}, GCD[#1, #2]] & @@ {#, k/#} &],
  {n, Length[s]}]

A376687 Numbers that set records in in A376281.

Original entry on oeis.org

24, 96, 120, 240, 360, 480, 840, 1080, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 10080, 15120, 27720, 30240, 55440, 60480, 83160, 110880, 151200, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 831600, 997920, 1330560, 1441440, 1663200, 2162160, 2882880
Offset: 1

Views

Author

Michael De Vlieger, Jan 08 2025

Keywords

Comments

Proper subset of the intersection A025487 and A379336.
There are three kinds of pairs (d, k/d), d | k, such that gcd(d, k/d) does not equal 1, d, or k/d:
Type A: rad(d) does not divide k/d, and rad(k/d) does not divide d (see A379752), where rad = A007947.
Type B: the squarefree kernel of one divisor divides the other but the reverse is not true (see A379772).
Type C: rad(d) = rad(k/d), i.e., d, k/d, and k are coreful (see A379552).
Conjecture: Numbers k that set records in A376281 do not have type C divisor pairs, i.e., those that are coreful but neither divides the other. This, since type C requires k to be powerful and divisible by cubes of 2 distinct primes (i.e., in A376936). Therefore the record is achieved only through large numbers of type A and B.
Since type A divisor pairs are common for composite k in A375055, this sequence is resembles A379752.
Since d and k/d are both composite, this sequence resembles A059992.
This sequence, to a lesser extent A379752, and a greater extent A059992, contains many highly composite numbers. (See plot of S(n) = union of this sequence and A002182 below, and corresponding graphs in respective other sequences.)

Examples

			Let b(n) = A376281(n).
Table showing exponents of prime power factors of a(n) for n = 1..20.
Example: a(5) = 360 = 2^3 * 3^2 * 5, hence we write "3.2.1".
   n    a(n)  Exp.   b(a(n))
  ----------------------------------
   1     24 *   3.1        1   4*6
   2     96     5.1        2   6*16 = 8*12
   3    120 **  3.1.1      3   4*30 = 6*20 = 10*12
   4    240 *   4.1.1      4   6*40 = 8*30 = 10*24 = 12*20
   5    360 **  3.2.1      5   4*90 = 10*36 = 12*30 = 15*24 = 18*20
   6    480     5.1.1      6   6*80 = 8*60 = 10*48 = 12*40 = 16*30 = 20*24
   7    840 *   3.1.1.1    7
   8   1080     3.3.1      9
   9   1680 *   4.1.1.1   10
  10   2160     4.3.1     11
  11   2520 **  3.2.1.1   13
  12   3360     5.1.1.1   14
*  = a(n) is highly composite (in A002182),
** = a(n) is superior highly composite (in both A002182 and A002201).

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..241
  • Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.
  • Michael De Vlieger, List of (d, k/d), d < k/d, k = a(n), n = 1..24, such that gcd(d, k/d) is not in {1, d, k/d}, showing type A in light gray, type B in either blue or gold, and type C (if it occurs) in black.
  • Michael De Vlieger, Prime power decomposition of a(n), n = 1..241.

Crossrefs

Cf. A002182, A002201, A007947, A025487, A059992, A379336, A376281, A377525, A379752.

Programs

  • Mathematica
    (* Load function f at A025487 *)
r = 0; s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not@*SquareFreeQ];
nn = Length[s]; Print[nn]
Reap[Monitor[
  Do[k = s[[i]];
    If[# > r, r = #; Sow[k]] &@
      Count[Transpose@{#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
        _?(And[1 < GCD @@ {##}, Mod[#1, #2] != 0,
           Mod[#2, #1] != 0] & @@ # &)], {i, nn}], i] ][[-1, 1]]

A380143 Sum of divisors d | k such that d and k/d share factors but both have a factor that does not divide the other, where k is in A375055.

Original entry on oeis.org

16, 20, 21, 48, 27, 28, 24, 25, 32, 60, 55, 39, 40, 32, 44, 45, 112, 65, 36, 84, 84, 52, 72, 35, 91, 57, 36, 96, 36, 140, 44, 63, 64, 45, 123, 40, 68, 108, 48, 85, 120, 75, 172, 96, 80, 136, 132, 56, 95, 48, 240, 49, 88, 48, 141, 92, 108, 93, 50, 196, 52, 172
Offset: 1

Views

Author

Michael De Vlieger, Jan 18 2025

Keywords

Comments

In other words, sum of divisors d | k such that gcd(d, k/d) > 1 but neither rad(d) | k/d nor rad(k/d) | d, where rad = A007947 and k is in A375055.
Define quality Q pertaining to 2 natural numbers a and b such that gcd(a, b) > 1 but neither rad(a) | b nor rad(b) | a.
Define function f(x) = A379752 to be the cardinality of divisor pairs (d, x/d) that have quality Q. f(x) > 0 for x in A375055, otherwise f(x) = 0.

Examples

			Let s = A375055.
a(1) = 16 since s(1) = 60 = 6*10; 6 + 10 = 16.
a(2) = 20 since s(2) = 84 = 6*14; 6 + 14 = 20.
a(3) = 21 since s(3) = 90 = 6*15; 6 + 15 = 21.
a(4) = 48 since s(4) = 120 = 6*20 = 10*12; 6 + 20 + 10 + 12 = 48, etc.

Links

Crossrefs

Cf. A007947, A375055, A379752.

Programs

  • Mathematica
    nn = 540; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Select[Range[nn], PrimeOmega[#] > PrimeNu[#] > 2 & ];
Table[k = s[[n]];
  DivisorSum[k, # &,
    And[1 < GCD @@ {##},
      Nor[Divisible[#2, rad[#1] ],
          Divisible[#1, rad[#2] ] ] ] & @@
    {#, k/#} &], {n, Length[s]}]
Showing 1-5 of 5 results.

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