



1, 2, 2, 8, 2, 144, 2, 64, 8, 1152, 2, 124416, 2, 1152, 144, 1024, 2, 35831808, 2, 221184, 144, 18432, 2, 859963392, 8, 18432, 64, 221184, 2, 261213880320000000, 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
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OFFSET

1,2


COMMENTS

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.
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.
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.
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.


LINKS

Antti Karttunen (terms 1..255) and Michael De Vlieger, Table of n, a(n) for n = 1..4619


FORMULA

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


EXAMPLE

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


MATHEMATICA

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 *)
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 *)


PROG

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


CROSSREFS

Cf. A046523, A162306, A243103, A283866, A283995.
Sequence in context: A056189 A331639 A121860 * A021442 A296851 A331750
Adjacent sequences: A283987 A283988 A283989 * A283991 A283992 A283993


KEYWORD

nonn


AUTHOR

Antti Karttunen and Michael De Vlieger, Mar 22 2017


STATUS

approved



