This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379593 #8 Jan 01 2025 01:47:54
%S A379593 8,32,128,512,2048,8192,20736,41472,82944,165888,186624,373248,746496,
%T A379593 1492992,2985984,5971968,6718464,11943936,23887872,26873856,53747712,
%U A379593 107495424,214990848,241864704,429981696,859963392,967458816,1719926784,3439853568,3869835264,7464960000
%N A379593 Numbers that set records in A379592.
%C A379593 Proper subset of the intersection of A025487 and A320966.
%C A379593 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).
%C A379593 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.)
%C A379593 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.
%H A379593 Michael De Vlieger, <a href="/A379593/b379593.txt">Table of n, a(n) for n = 1..119</a>
%H A379593 Michael De Vlieger, <a href="/A379593/a379593.txt">Prime power decomposition of a(n)</a>, n = 1..119.
%e A379593 Let b(n) = A379592(n).
%e A379593 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.
%e A379593 n a(n) exp. b(a(n))
%e A379593 --------------------------
%e A379593 1 8 3 1 2*4
%e A379593 2 32 5 2 2*16 = 4*8
%e A379593 3 128 7 3 2*64 = 4*32 = 8*16
%e A379593 4 512 9 4 2*256 = 4*128 = 8*64 = 16*32
%e A379593 5 2048 11 5 2*1024 = 4*512 = 8*256 = 16*128 = 32=64
%e A379593 6 8192 13 6 2*4096 = 4*2048 = 8*1024 = 16*512 = 32*256 = 64*128
%e A379593 7 20736 8.4 7
%e A379593 8 41472 9.4 8
%e A379593 9 82944 10.4 9
%e A379593 10 165888 11.4 10
%e A379593 11 186624 8.6 11
%e A379593 12 373248 9.6 12
%t A379593 (* Load function f at A025487 *)
%t A379593 r = 0; s = Union@ Flatten@ f[10]; nn = Length[s];
%t A379593 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A379593 Transpose@ Reap[Monitor[
%t A379593 Do[k = s[[i]];
%t A379593 If[# > r, r = #; Sow[k]] &@
%t A379593 Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
%t A379593 _?(And[rad[#1] == rad[#2],
%t A379593 Xor[Divisible[#2, #1],
%t A379593 Divisible[#1, #2]]] & @@ # &)], {i, nn}], {i, nn}] ][[-1, 1]]
%Y A379593 Cf. A005934, A025487, A320966, A361430, A370329, A379592, A379594.
%K A379593 nonn
%O A379593 1,1
%A A379593 _Michael De Vlieger_, Dec 30 2024