A379593 Numbers that set records in A379592.

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

Author

Michael De Vlieger, Dec 30 2024

Keywords

• nonn

Comments

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.

Examples

			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
		

Links

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

Crossrefs

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

Programs

Mathematica

(* 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]]