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A283990 a(n) = A046523(A243103(n)). 3

%I

%S 1,2,2,8,2,144,2,64,8,1152,2,124416,2,1152,144,1024,2,35831808,2,

%T 221184,144,18432,2,859963392,8,18432,64,221184,2,261213880320000000,

%U 2,32768,1152,589824,144,26748301344768,2,589824,1152,1528823808,2,12036735605145600000,2,7077888,124416,589824,2,1283918464548864,8,27518828544,1152,7077888,2,69331597085638656,144

%N a(n) = A046523(A243103(n)).

%C Irregular triangle A162306(n) lists numbers 1 <= m <= n with m | n^e and e >= 0 of which A243103(n) is the product. The numbers m in this range have prime factors p that also divide n, with 1 the empty product that divides n; a(1) = 1 because 1 is the empty product and has no prime factors.

%C For prime p, a(p) = 2 since A162306(p) = {1,p}, the product of which is p. The smallest number with prime signature of a prime is the smallest prime, 2.

%C For prime power n = p^e, a(p^e) = 2^A000217(e), since A162306(p^e) is the power range 0..e of p | n.

%C The sequence can also be produced by raising the multiplicities m listed in A283866(n) = T(n,k) to the prime corresponding to their order in row n, i.e., prime(k)^m, and taking the product.

%H Antti Karttunen (terms 1..255) and Michael De Vlieger, <a href="/A283990/b283990.txt">Table of n, a(n) for n = 1..4619</a>

%F a(n) = A046523(A243103(n)).

%e a(9) = 8 since A162306(9) = {1,3,9}, the product of which is 27 = 3^3; 2 is the smallest prime, thus a(9) = 2^3 = 8.

%e a(15) = 144 since A162306(15) = {1,3,5,9,15}, the product of which is 2025 = 3^4 * 5^2; applying these multiplicities to the smallest primes in their order gives us 2^4 * 3^2 = 144.

%e a(4620) = 2^908 * 3^511 * 5^299 * 7^220 * 11^155 has 1072 decimal digits.

%t Table[If[n == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, FactorInteger[Apply[Times, Select[Range@ n, PowerMod[n, Floor@ Log2@ n, #] == 0 &]]][[All, -1]]]], {n, 55}] (* or *)

%t Table[Times @@ MapIndexed[Prime[First@ #2]^#1 &, With[{m = Floor@ Log2@ n}, Values@ Merge[Association /@ Map[#1 -> #2 & @@ # &, FactorInteger@ Rest@ Select[Range@ n, PowerMod[n, m, #] == 0 &], {2}], Total]] /. {} -> {0}], {n, 10^4}] (* _Michael De Vlieger_, Mar 22 2017, Version 10, faster *)

%o (Scheme) (define (A283990 n) (A046523 (A243103 n)))

%Y Cf. A046523, A162306, A243103, A283866, A283995.

%K nonn

%O 1,2

%A _Antti Karttunen_ and _Michael De Vlieger_, Mar 22 2017

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Last modified May 27 13:00 EDT 2020. Contains 334657 sequences. (Running on oeis4.)