A378767 -id:A378767 - cp's OEIS frontend

A379772 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d} and either rad(d) | k/d or rad(k/d) | d, where k = A378767(n).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Michael De Vlieger, Jan 02 2025

nonn

Let rad = A007947 and let omega = A001221.

Number of ways to write k = A378767(n) as a product of numbers i and j, omega(i) < omega(j) = omega(i*j), that are neither coprime nor divide one another, where rad(i) | j, but rad(j) does not divide i. Both i and j are necessarily composite.

			Let s(n) = A378767(n).
a(1) = 1 since s(1) = 24 = 4*6, omega(4) < omega(6) = omega(24), rad(4) | 6.
a(2) = 1 since s(2) = 40 = 4*10, omega(4) < omega(10) = omega(40), rad(4) | 10.
a(3) = 1 since s(3) = 48 = 6*8, omega(8) < omega(6) = omega(48), rad(8) | 6.
a(9) = 2 since s(9) = 96 = 6*16 = 8*12.
a(54) = 3 since s(54) = 384 = 6*64 = 12*32 = 16*24.
a(165) = 5 since s(165) = 1080 = 4*270 = 9*120 = 12*90 = 18*60 = 30*36.

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

Cf. A001221, A007947, A378767.

nn = 120; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Select[Select[Range[nn],
  AnyTrue[FactorInteger[#][[All, -1]], # > 2 &] &],
    Not @* PrimePowerQ];
Table[k = s[[n]];
  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]]) & @@ # &)], {n, Length[s]}]

A379336 Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

Original entry on oeis.org

24, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272, 276, 280, 288, 294

Offset: 1

Michael De Vlieger, Dec 24 2024

Both divisors d and d/k are composite, since primes p either divide or are coprime to another number, and all numbers smaller than p are coprime to p.
Proper subset of A126706; contains A378769, which in turn contains A378984.
Consider composite k, m, k != m. Define a "neutral" relation to be such that 1 < gcd(k,m) and not equal to either k or m. Then neither k nor m divides the other, and k and m are not coprime. If k is neutral to m, then m is neutral to k, since order does not matter. Then either the squarefree kernel of one divides the other or it does not. Thus, there are 3 kinds of neutral relation:
Type A: Though gcd(k,m) > 1, k has a factor P that does not divide m, and m has a factor Q that does not divide k.
Type B: rad(k) = rad(m), yet neither k divides m nor m divides k, where rad = A007947 is the squarefree kernel.
Type C: Squarefree kernel of one number divides the other, while the other has a factor that does not divide the former.
A378769, subset of this sequence, contains numbers k that have all 3 types of neutral relation between at least 1 divisor pair (d, k/d) for each.

			a(1) = 24 = 2^3 * 3 = 4*6, both composite; gcd(4,6) = 2, 4 does not divide 6 (type C).
a(2) = 40 = 2^3 * 5 = 4*10, gcd(4,10) = 2 (type C).
a(3) = 48 = 2^4 * 3 = 6*8, gcd(6,8) = 2 (type C).
a(6) = 60 = 2^2 * 3 * 5 = 6*10, gcd(6,10) = 2 (type A).
a(12) = 96 = 2^5 * 3 = 6*16 = 8*12, both type C.
a(38) = 216 = 2^3 * 3^3 = 4*54 (type C) = 9*24 (type C) = 12*18 (type B)
a(1605) = 5400 = 2^3 * 3^3 * 5^2 = 4*1350 (type C) = 24*225 (type A) = 60*90 (type B) = A378769(1).
a(10475) = 32400 = 2^4 * 3^4 * 5^2 = 8*4050 (type C) = 48*675 (type A) = 120*270 (type B) = A378984(1) = A378769(14), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Chart of (d, a(n)/d) for a(n) = 1..144, showing only the smallest d for each type of neutral relation, where type A is shown in gray, type B in black, and type C in either blue or gold.

Cf. A045763, A085986, A126706, A375055, A376271, A376936, A378767, A378769, A378984.

nn = 300; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
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 ]

This sequence is A376271 \ A085986 = {k : bigomega(k) > omega(k) > 1, bigomega(k) > 3} \ { k^2 : bigomega(k) = omega(k) = 2 }, where bigomega = A001222 and omega = A001221.

Union of A375055, A376936, and A378767.

A378769 Intersection of A375055 and A376936.

Original entry on oeis.org

5400, 9000, 10584, 10800, 13500, 16200, 18000, 21168, 21600, 24696, 26136, 27000, 31752, 32400, 36000, 36504, 37044, 40500, 42336, 43200, 45000, 48600, 49000, 49392, 52272, 54000, 62424, 63504, 64800, 67500, 68600, 72000, 73008, 74088, 77976, 78408, 81000, 84672

Offset: 1

Michael De Vlieger, Dec 13 2024

Let omega = A001221, bigomega = A001222, rad = A007947.
Powerful numbers k with bigomega(k) > omega(k) > 2 that are divisible by two distinct prime cubes p^3 and q^3.
Numbers k such that there exists (d, k/d), d | k, such that d neither divides nor is coprime to k/d and vice versa in the following 3 ways:
Type A: rad(d) does not divide d/k and rad(d/k) does not divide d
Type B: rad(d) divides d/k but rad(d/k) does not divide d
Type C: rad(d) | d/k and rad(d/k) | d, hence rad(d) = rad(d/k) = rad(k), a kind of coreful divisor pair.
Since (d, d/k) are noncoprime and do not divide one another, both must be composite, thus k is also composite.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.
A378767 = { k : omega(k) > 1, p^3 | k for some prime p }, and
A376936 = { k : rad(k)^2 | k, p^3 | k and q^3 | k for distinct primes p, q }.
Therefore, we need only take intersection of A375055 and A376936.

			Table of the first 12 terms of this sequence, showing examples of types A, B, and C described in Comments.
   n     a(n)  Factors of a(n)    Type A      Type B      Type C
  ----------------------------------------------------------------
   1    5400   2^3 * 3^3 * 5^2    24 * 225    4 * 1350    60 * 90
   2    9000   2^3 * 3^2 * 5^3    18 * 500    4 * 2250    60 * 150
   3   10584   2^3 * 3^3 * 7^2    24 * 441    4 * 2646    84 * 126
   4   10800   2^4 * 3^3 * 5^2    48 * 225    8 * 1350    90 * 120
   5   13500   2^2 * 3^3 * 5^3    12 * 1125   9 * 1500    90 * 150
   6   16200   2^3 * 3^4 * 5^2    24 * 675    4 * 4050    60 * 270
   7   18000   2^4 * 3^2 * 5^3    18 * 1000   8 * 2250   120 * 150
   8   21168   2^4 * 3^3 * 7^2    48 * 441    8 * 2646   126 * 168
   9   21600   2^5 * 3^3 * 5^2    50 * 432    8 * 2700    90 * 240
  10   24696   2^3 * 3^2 * 7^3    18 * 1372   4 * 6174    84 * 294
  11   26136   2^3 * 3^3 * 11^2   24 * 1089   4 * 6534   132 * 198
  12   27000   2^3 * 3^3 * 5^3    24 * 1125   4 * 6750    60 * 450

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..16, showing divisor pairs of type A in light gray, type B in blue and gold, and type C in black.

Cf. A001694, A024619, A126706, A375055, A376936, A378767, A378900.

Cf. A082020, A082695.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^16],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Select[s, PrimeOmega[#] > PrimeNu[#] > 2 &]

Intersection of A375055, A376936, and A378767.
This sequence is { k : rad(k)^2 | k, bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) - ((Sum_{p prime} (1/(p^2*(p-1))))^2 - Sum_{p prime} (1/(p^4*(p-1)^2)))/2 = 0.0025524144364532126894... . - Amiram Eldar, Dec 21 2024

A378900 Squares of numbers divisible by the squares of two distinct primes.

Original entry on oeis.org

1296, 5184, 10000, 11664, 20736, 32400, 38416, 40000, 46656, 50625, 63504, 82944, 90000, 104976, 129600, 153664, 156816, 160000, 186624, 194481, 202500, 219024, 234256, 250000, 254016, 291600, 331776, 345744, 360000, 374544, 419904, 455625, 456976, 467856, 490000

Offset: 1

Michael De Vlieger, Dec 12 2024

Also, the squares in A376936.
Proper subset of A378767, in turn a proper subset of A286708, the intersection of A001694 and A024619.
Numbers that have 3 kinds of coreful divisor pairs (d, k/d), d | k, i.e., rad(d) = rad(k/d) = rad(k) where rad = A007947. These kinds are described as follows:
Type A: d = k/d, which pertain to square k (in A000290).
Type B: d | k/d, d < k/d, which pertain to k in A320966, powerful numbers divisible by a cube.
Type C: neither d | k/d nor k/d | d, which pertain to k in A376936.
Since divisors d, k/d may either divide or not divide the other, there are no other cases.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.

			Let b = A036785.
Table of the first 12 terms of this sequence, showing examples of types A, B, and C of coreful pairs of divisors.
   n    a(n)   Factors of a(n)    b(n)   Type B       Type C
  -------------------------------------------------------------
   1    1296   2^4  * 3^4          36    6 * 216      24 * 54
   2    5184   2^6  * 3^4          72    6 * 864      48 * 108
   3   10000   2^4  * 5^4         100   10 * 1000     40 * 250
   4   11664   2^4  * 3^6         108    6 * 1944     24 * 486
   5   20736   2^8  * 3^4         144    6 * 3456     54 * 384
   6   32400   2^4  * 3^4 * 5^2   180   30 * 1080    120 * 270
   7   38416   2^4  * 7^4         196   14 * 2744     56 * 686
   8   40000   2^6  * 5^4         200   10 * 4000     80 * 500
   9   46656   2^6  * 3^6         216    6 * 7776     48 * 972
  10   50625   3^4  * 5^4         225   15 * 3375    135 * 375
  11   63504   2^4  * 3^4 * 7^2   252   42 * 1512    168 * 378
  12   82944   2^10 * 3^4         288    6 * 13824    54 * 1536

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..12, showing divisor pairs of type A in light gray, type B in orange and purple, and type C in black.

Cf. A000290, A001694, A007947, A024619, A036785, A126706, A286708, A320966, A376936, A378768.

Cf. A013661, A082020.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}],   IntegerQ@ Sqrt[#] &] &[500000];
Union@ Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

a(n) = A036785(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} 1/(p^4-1)) = 0.0015294876575980711757... . - Amiram Eldar, Dec 21 2024

A378984 Squares in A378769.

Original entry on oeis.org

32400, 63504, 90000, 129600, 156816, 202500, 219024, 254016, 291600, 345744, 360000, 374544, 467856, 490000, 518400, 571536, 627264, 685584, 777924, 810000, 876096, 960400, 1016064, 1089936, 1166400, 1210000, 1245456, 1382976, 1411344, 1440000, 1498176, 1587600

Offset: 1

Michael De Vlieger, Dec 15 2024

Let omega = A001221, bigomega = A001222, and rad = A007947.
Numbers k that have all types of divisor pairs (d, k/d), d | k, that are listed in both A378769 and A378900. These are listed below:
Type A*: (Nontrivial) unitary divisor pairs, i.e., d coprime to k/d. The rest of the types are in cototient.
Type B*: gcd(d, k/d) > 1, rad(d) !| k/d, rad(k/d) !| d. These exist for k in A375055.
Type C: d < k/d, d | k/d but rad(k/d) !| d. Implies rad(k/d) = rad(k) and omega(d) < omega(k/d). These exist for k in A126706.
Type D: Either rad(d) | k/d, rad(k/d) !| d or vice versa. These exist for k in A378767.
Type E*: d = k/d = sqrt(k).
Type F: rad(d) = rad(k/d) = rad(k), d < k/d, d | k/d. These exist for k in A320966.
Type G*: rad(d) = rad(k/d) = rad(k), neither d | k/d nor k/d | d. These exist for k in A376936.
Asterisks denote symmetric types.
Since numbers d and k/d are either coprime or not, and if not, the squarefree kernel of one either divides the other or not, and if so, d divides k/d or not, and if so, d = k/d or not, there are no other types.
Smallest odd term is a(45) = 2480625.
Square roots not A350372: sqrt(810000) = 900 is not in A350372.

			a(1) = 32400 = 2^4 * 3^4 * 5^2 has the following divisor pair types:
  Type A: 16 * 2025, Type B: 48 * 675, Type C: 2 * 16200, Type D: 8 * 4050
  Type E: 180 * 180, Type F: 30 * 1080, Type G: 120 * 270.
a(2) = 63504 = 2^4 * 3^4 * 7^2 has the following divisor pair types:
  Type A: 16 * 3969, Type B: 48 * 1323, Type C: 2 * 31752, Type D: 8 * 7938
  Type E: 252 * 252, Type F: 42 * 1512, Type G: 168 * 378.
a(3) = 90000 = 2^4 * 3^2 * 5^4 has the following divisor pair types:
  Type A: 9 * 10000, Type B: 18 * 5000, Type C: 2 * 45000, Type D: 8 * 11250
  Type E: 300 * 300, Type F: 30 * 3000, Type G: 120 * 750, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram listing all divisor pairs for a(n), n = 1..8, showing type A in white, type B in light gray, type C in green or red, type D in blue or gold, type E in dark gray, type F in orange or purple, and type G in black.
Michael De Vlieger, Diagram listing divisor pairs (d, k/d) for k = a(n), n = 1..60, showing only those with the smallest d and using the same color scheme as above, for each type and its reversal if the type is nonsymmetric.

Cf. A000290, A001221, A001222, A007947, A126706, A320966, A350372, A375055, A376936, A378767, A378769, A378900.

Cf. A013661, A082020.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^21],  IntegerQ@ Sqrt[#] &];
t = Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Select[t, PrimeOmega[#] > PrimeNu[#] > 2 &]

This sequence is { k = s^2 : rad(k)^2 | k,
  bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
Intersection of A378769 and A378900.
Intersection of A000290, A375055, and A376936.
Sum_{n>=1} = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} (1/(p^4-1))) - ((Sum_{p prime} (1/(p^2*(p^2-1))))^2 - Sum_{p prime} (1/(p^4*(p^2-1)^2)))/2 = 0.00015490158528995570146... . - Amiram Eldar, Dec 21 2024

A379773 Numbers that set records in in A379772.

Original entry on oeis.org

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]]

cp's OEIS Frontend

A379772 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d} and either rad(d) | k/d or rad(k/d) | d, where k = A378767(n).

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A379336 Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

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A378769 Intersection of A375055 and A376936.

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A378900 Squares of numbers divisible by the squares of two distinct primes.

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A378984 Squares in A378769.

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A379773 Numbers that set records in in A379772.

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A379774 Records in A379772.

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