|
| |
|
|
A238745
|
|
a(1) = 1; for n > 1, if the first integer with the same prime signature as n is factorized into primorials as Product A002110(i)^e(i), then a(n) = Product prime(i)^e(i).
|
|
20
|
|
|
|
1, 2, 2, 4, 2, 3, 2, 8, 4, 3, 2, 6, 2, 3, 3, 16, 2, 6, 2, 6, 3, 3, 2, 12, 4, 3, 8, 6, 2, 5, 2, 32, 3, 3, 3, 9, 2, 3, 3, 12, 2, 5, 2, 6, 6, 3, 2, 24, 4, 6, 3, 6, 2, 12, 3, 12, 3, 3, 2, 10, 2, 3, 6, 64, 3, 5, 2, 6, 3, 5, 2, 18, 2, 3, 6, 6, 3, 5, 2, 24, 16, 3, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Alternate definition: a(1) = 1; for n > 1, if row n of table A238744 is {k(1), k(2),...,k(A051903(n))}, then a(n) = Product {i = 1 to A051903(n)} prime(k(i)).
Since the first integer of each prime signature (A025487) is always a product of primorials (A002110), there is always a value for a(n). Every positive integer appears in the sequence.
a(m) = a(n) iff m and n have the same prime signature. If the prime signatures of m and n are conjugate to each other when they are viewed as partitions, then a(n) = A181819(m) and a(m) = A181819(n).
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
The first integer with the same prime signature as 40 is 24 = 2^3*3. Since the factorization of 24 into primorials is 24 = 2^2*6 = A002110(1)^2*A002110(2), a(24) = a(40) = prime(1)^2*prime(2) = 2^2*3 = 12.
|
|
|
MATHEMATICA
|
f[n_] := Block[{k = 1, d, a}, While[n - Times @@ Prime@ Range[k + 1] >= 0, k++]; If[n == Product[Prime@ i, {i, k}], Prime@ k, d = Select[Reverse@ FoldList[#1 #2 &, Prime@ Range@ k], Divisible[n, #] &]; If[AllTrue[#, IntegerQ], Times @@ Map[(FactorInteger[#1][[-1, 1]])^#2 & @@ # &, Reverse@ Tally@ #], False] &@ Rest@ NestWhileList[Function[P, {#1/P, P}]@ SelectFirst[d, Function[k, Divisible[#1, k]]] & @@ # &, {n, 1}, First@ # > 1 &][[All, -1]]]]; Table[f@ Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]] - Boole[n == 1], {n, 83}] (* Michael De Vlieger, May 19 2017, Version 10.2 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
