A379772 -id:A379772 - cp's OEIS frontend

A379773 Numbers that set records in in A379772.

24, 96, 384, 1080, 2160, 4320, 8640, 12960, 17280, 34560, 38880, 69120, 77760, 108000, 155520, 311040, 432000, 622080, 756000, 1512000, 2268000, 3024000, 4536000, 5292000, 6804000, 9072000, 12096000, 13608000, 21168000, 27216000, 47628000, 54432000, 74088000, 81648000

Offset: 1

Michael De Vlieger, Jan 04 2025

nonn

Proper subset of the intersection A025487 and A378767.

Conjecture: a(n) is powerful (i.e., in A286708) for n >= 68. Additionally, for some n much larger than 68, a(n) may be cubefull (i.e., in A372695).

			Let b(n) = A379772(n).
Table showing exponents of prime power factors of a(n) for n = 1..20.
Example: a(5) = 2160 = 2^4 * 3^3 * 5, hence we write "4.3.1".
   n     a(n)  Exp.   b(a(n))
  ----------------------------------
   1      24   3.1      1   4*6
   2      96   5.1      2   6*16 = 8*12
   3     384   7.1      3   6*64 = 12*32 = 16*24
   4    1080   3.3.1    5   4*270 = 9*120 = 12*90 = 18*60 = 30*36
   5    2160   4.3.1    6   8*270 = 9*240 = 18*120 = 24*90 = 30*72 = 36*60
   6    4320   5.3.1    9
   7    8640   6.3.1   10
   8   12960   5.4.1   11
   9   17280   7.3.1   13
  10   34560   8.3.1   14
  11   38880   5.5.1   16
  12   69120   9.3.1   17

Michael De Vlieger, Table of n, a(n) for n = 1..171
Michael De Vlieger, Prime power decomposition of a(n), n = 1..171.
Michael De Vlieger, List of (d, k/d), d < k/d, k = a(n), n = 1..24, such that gcd(d, k/d) > 1, and shown in blue, rad(d) | k/d though d does not divide k/d, but rad(k/d) does not divide d, while in gold, rad(d) does not divide k/d but rad(k/d) | d though k/d does not divide d.

Cf. A025487, A378767, A379772, A379774.

(* Load function f at A025487 *)
r = 0; s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not @* SquareFreeQ];
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; nn = Length[s];
Reap[Do[k = s[[i]];
  If[# > r, r = #; Sow[k] ] &@
    Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
      _?((m = GCD @@ {##};
      And[! MemberQ[{1, #1, #2}, m],
        And[PrimeNu[#1] < PrimeNu[#2], Divisible[#2, rad[#1]]] & @@
        SortBy[{##}, PrimeNu]]) & @@ # &)], {i, nn}] ][[-1, 1]]

A379774 Records in A379772.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 11, 13, 14, 16, 17, 18, 19, 23, 25, 26, 30, 33, 38, 42, 47, 48, 52, 57, 60, 61, 66, 73, 81, 86, 90, 93, 94, 98, 105, 112, 120, 124, 129, 132, 138, 143, 148, 154, 155, 177, 196, 203, 204, 225, 228, 244, 267, 269, 273, 282, 300, 318, 342, 345

Offset: 1

Michael De Vlieger, Jan 04 2025

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..171

Cf. A378767, A379772, A379773.

(* Load function f at A025487 *)
r = 0; s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not @* SquareFreeQ];
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; nn = Length[s];
Reap[Do[k = s[[i]];
  If[# > r, r = #; Sow[r] ] &@
    Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
      _?((m = GCD @@ {##};
      And[! MemberQ[{1, #1, #2}, m],
        And[PrimeNu[#1] < PrimeNu[#2], Divisible[#2, rad[#1]]] & @@
        SortBy[{##}, PrimeNu]]) & @@ # &)], {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

Michael De Vlieger, Jan 08 2025

nonn

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.

			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.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

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

Michael De Vlieger, Jan 08 2025

nonn

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.)

			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).

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.

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

(* 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]]

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A379773 Numbers that set records in in A379772.

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A379774 Records in A379772.

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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.

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A376687 Numbers that set records in in A376281.

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