A370329 -id:A370329 - cp's OEIS frontend

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

216, 432, 648, 864, 1000, 1296, 1728, 1944, 2000, 2592, 2744, 3375, 3456, 3888, 4000, 5000, 5184, 5400, 5488, 5832, 6912, 7776, 8000, 9000, 9261, 10000, 10125, 10368, 10584, 10648, 10800, 10976, 11664, 13500, 13824, 15552, 16000, 16200, 16875, 17496, 17576, 18000

Offset: 1

Michael De Vlieger, Oct 16 2024

Numbers m with coreful divisors d, m/d such that neither d | m/d nor m/d | d, i.e., numbers m such that there exists a divisor pair (d, m/d) such that rad(d) = rad(m/d) but gcd(d, m/d) > 1 is neither d nor m/d, where rad = A007947. Divisors in each pair must be dissimilar and each in A126706.
Proper subset of A320966.
Contains A372695, A177493, and A162142. Does not contain A085986.

			216 is in the sequence since rad(12) | rad(18), but 12 does not divide 18 and 18 does not divide 12.
432 is a term since rad(18) | rad(24), but 18 does not divide 24 and 24 does not divide 18.
Table of coreful divisors d, a(n)/d such that neither d | a(n)/d nor a(n)/d | d for select a(n)
   n |   a(n)   divisor pairs d X a(n)/d
  ---------------------------------------------------------------------------
   1 |   216:   12 X 18;
   2 |   432:   18 X 24;
   3 |   648:   12 X 54;
   4 |   864:   24 X 36, 18 X 48;
   5 |  1000:   20 X 50;
   6 |  1296:   24 X 54;
   7 |  1728:   18 X 96, 36 X 48;
   8 |  1944:   12 X 162, 36 X 54;
   9 |  2000:   40 X 50;
  10 |  2592:   24 X 108, 48 X 54;
  11 |  2744:   28 X 98;
  12 |  3375:   45 X 75;
  13 |  3456:   18 X 192, 36 X 96, 48 X 72;
  22 |  7776:   24 X 324, 48 X 162, 54 X 144, 72 X 108;
  58 | 31104:   48 X 648, 54 X 576, 96 X 324, 108 X 288, 144 X 216, 162 X 192

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Notes on this sequence

Cf. A007947, A030078, A085986, A126706, A162142, A177493, A286708, A307958, A320966, A361430, A370329, A372695, A376514.

Cf. A082020, A082695.

Union@ Select[
  Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[20000],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) = 0.021194288968234037106579437374641326044... . - Amiram Eldar, Nov 08 2024

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]}]

A370328 The number of powerful divisors of the powerful numbers.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 6, 4, 4, 6, 2, 3, 7, 8, 2, 4, 6, 9, 4, 5, 8, 10, 2, 8, 3, 2, 6, 8, 12, 4, 4, 6, 9, 2, 12, 4, 12, 6, 4, 6, 8, 10, 2, 15, 8, 2, 6, 10, 9, 10, 4, 6, 14, 4, 4, 16, 6, 3, 6, 2, 6, 4, 4, 10, 12, 2, 18, 8, 12, 2, 8, 15, 12, 8, 11, 4, 7

Offset: 1

Amiram Eldar, Feb 15 2024

The product of the exponents of the prime factorization of the powerful numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001694, A005361, A306458, A363194, A370329.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1 || Min[e] > 1, Times @@ e, Nothing]]; Array[f, 2500]

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1 || vecmin(e) > 1, print1(vecprod(e), ", ")));}

a(n) = A005361(A001694(n)).

a(n) = A000005(A306458(n)).

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]]

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A376936 Powerful numbers divisible by cubes of 2 distinct primes.

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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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A370328 The number of powerful divisors of the powerful numbers.

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A379593 Numbers that set records in A379592.

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