A379552 -id:A379552 - cp's OEIS frontend

A379554 Records in A379552.

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 18, 20, 21, 24, 26, 27, 28, 30, 32, 36, 40, 42, 48, 52, 54, 56, 60, 64, 72, 78, 80, 90, 96, 100, 108, 114, 120, 126, 128, 135, 136, 144, 150, 160, 168, 170, 176, 180, 186, 192, 200, 204, 210, 224, 240, 248, 252, 264, 272, 280, 288

Offset: 1

Michael De Vlieger, Dec 25 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..212

Cf. A376936, A379552, A379553.

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[r]] &@
  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]]

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

Michael De Vlieger, Dec 25 2024

nonn

Proper subset of A025487.

			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.

Michael De Vlieger, Table of n, a(n) for n = 1..212 (a(n) <= A002110(19).)
Michael De Vlieger, Prime power decomposition of a(n), n = 1..212.

Cf. A376936, A379552, A379554.

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]]

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]}]

A379592 Number of coreful divisor pairs (d, k/d), d | k, d < k/d, such that only one divisor divides the other, where k is in A320966.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 1, 4, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 4, 1, 3, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 1, 5, 3, 1, 3, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 1, 3, 1, 1, 2, 4, 1, 2, 5, 1, 1, 1, 4, 1, 1, 2, 5, 1, 1

Offset: 1

Michael De Vlieger, Dec 28 2024

nonn

Number of ways to write k = A320966(n) as a product of numbers i and j, i < j, such that rad(i) = rad(j) = rad(k), and either i | j or j | i, where rad = A007947 is the squarefree kernel.

Analogous to A370329, where the reference domain is A001694 instead of A320966.

			Let s(n) = A320966(n).
a(1) = 1 since s(1) = 8 = 2*4.
a(2) = 1 since s(2) = 16 = 2*8.
a(3) = 1 since s(3) = 27 = 3*9.
a(4) = 2 since s(4) = 32 = 2*16 = 4*8.
a(10) = 3 since s(10) = 128 = 2*64 = 4*32 = 8*16.
a(23) = 4 since s(23) = 512 = 2*256 = 4*128 = 8*64 = 16*32.
a(181) = 7 since s(181) = 20736 = 6*3456 = 12*1728 = 18*1152 = 24*864 = 36*576 = 48*432 = 72*288, etc.

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

Cf. A320966, A370329, A379552.

nn = 5400; 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 &] > 0 &];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
    _?(And[rad[#1] == rad[#2],
       Xor[Divisible[#2, #1],
           Divisible[#1, #2]]] & @@ # &)], {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]]

A379593 Numbers that set records in A379592.

Original entry on oeis.org

8, 32, 128, 512, 2048, 8192, 20736, 41472, 82944, 165888, 186624, 373248, 746496, 1492992, 2985984, 5971968, 6718464, 11943936, 23887872, 26873856, 53747712, 107495424, 214990848, 241864704, 429981696, 859963392, 967458816, 1719926784, 3439853568, 3869835264, 7464960000

Offset: 1

Michael De Vlieger, Dec 30 2024

nonn

Proper subset of the intersection of A025487 and A320966.
Let k be a powerful number (in A001694) and let coreful d | k be such that k/d is also coreful, i.e., rad(d) = rad(d/k) = rad(k), where rad = A007947 is the squarefree kernel. Suppose d < d/k. Then coreful d may either divide k/d or not (indeed, if d/k < d, k/d may either divide d or not).
Then we have either d | k/d (the cardinality of such divisors is A379592(n) for k = A320966(n)) or d does not divide k/d (the cardinality of such divisors is A379552(n) for k = A376936(n)). (The case d = k/d, both certainly coreful, of course pertains to perfect squares k in A000290.)
Coreful divisors are counted by A361430 across natural numbers, and A370329 across powerful numbers A001694. Numbers that set records in A361430 (and A370329) are in A005934 (highly powerful numbers), with records in A036965.

			Let b(n) = A379592(n).
Table showing exponents of prime power factors of a(n) for n = 1..12. Example: a(7) = 20736 = 2^8*3^4, so "8.4" appears in the "exp." column.
   n      a(n)  exp. b(a(n))
  --------------------------
   1        8    3       1   2*4
   2       32    5       2   2*16 = 4*8
   3      128    7       3   2*64 = 4*32 = 8*16
   4      512    9       4   2*256 = 4*128 = 8*64 = 16*32
   5     2048   11       5   2*1024 = 4*512 = 8*256 = 16*128 = 32=64
   6     8192   13       6   2*4096 = 4*2048 = 8*1024 = 16*512 = 32*256 = 64*128
   7    20736    8.4     7
   8    41472    9.4     8
   9    82944   10.4     9
  10   165888   11.4    10
  11   186624    8.6    11
  12   373248    9.6    12

Michael De Vlieger, Table of n, a(n) for n = 1..119
Michael De Vlieger, Prime power decomposition of a(n), n = 1..119.

Cf. A005934, A025487, A320966, A361430, A370329, A379592, A379594.

(* 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[k]] &@
        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]]

A379465 Sum of coreful divisors d | k such that gcd(d, k/d) is not in {1, d, k/d} and rad(d) = rad(k/d), where k is in A376936 and rad = A007947.

Original entry on oeis.org

30, 42, 66, 126, 70, 78, 198, 264, 90, 234, 126, 120, 462, 312, 270, 270, 306, 150, 154, 696, 798, 936, 390, 210, 210, 290, 210, 714, 210, 286, 210, 462, 744, 240, 1710, 1224, 910, 330, 420, 2262, 390, 270, 714, 870, 1050, 294, 330, 630, 630, 2232, 378, 1620, 330

Offset: 1

Michael De Vlieger, Jan 13 2025

nonn

Define function f(x) to be 1/2 * card({ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) }), a counting function of distinct divisor pairs (d, x/d) that are both coreful but neither divides the other.
Define function g(x) to be Sum_{ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) } d.
Define h(x) to be equal to A364988(x) = Sum_{ d | x : rad(d) = rad(x) } d, sum of coreful divisors of x.
a(n) = g(A376936(n)) <= h(A376936(n)).
The function f(x) is analogous to tau(x) = A000005(x) while g(x) is analogous to sigma(x) = A000203(x).
f(k) > 0 and g(k) > 1 for k in A376936, otherwise f(k) = 0 and g(k) = 0.
Since rad(d) = rad(k/d) = rad(k), a(n) = m*rad(k), with integer m > 1.

			Let s = A376936.
a(1) = 30 since s(1) = 216 = 12*18 = 2*6 + 3*6 = 5*rad(216), and the sum of these is 30.
a(2) = 42 since s(2) = 432 = 18*24 = 3*6 + 4*6 = 7*rad(432), and the sum of these is 42.
a(3) = 66 since s(3) = 648 = 12*54 = 2*6 + 9*6 = 11*rad(648), and their sum is 66.
a(4) = 126 since s(4) = 864 = 18*48 = 24*36, and the sum of all these divisors is 126, etc. Note that 18 + 48 + 24 + 36 = 3*6 + 8*6 + 4*6 + 6*6 = 21*rad(864).

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

Cf. A364988, A379552.

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 &];
Map[Function[n,
  DivisorSum[n, # &,
    And[! MemberQ[{1, #1, #2}, GCD @@ {##}],
      rad[#1] == rad[#2]] & @@ {#, n/#} &]], s]

cp's OEIS Frontend

A379554 Records in A379552.

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A379553 Numbers k in A376936 that set records in A379552.

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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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A379592 Number of coreful divisor pairs (d, k/d), d | k, d < k/d, such that only one divisor divides the other, where k is in A320966.

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A376687 Numbers that set records in in A376281.

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A379593 Numbers that set records in A379592.

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A379465 Sum of coreful divisors d | k such that gcd(d, k/d) is not in {1, d, k/d} and rad(d) = rad(k/d), where k is in A376936 and rad = A007947.

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