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A278245
Least number with the same prime signature as the n-th Fibonacci number: a(n) = A046523(A000045(n)).
8
1, 1, 2, 2, 2, 8, 2, 6, 6, 6, 2, 144, 2, 6, 30, 30, 2, 120, 6, 210, 30, 6, 2, 10080, 12, 6, 210, 210, 2, 9240, 6, 210, 30, 6, 30, 166320, 30, 30, 30, 30030, 6, 9240, 2, 2310, 2310, 30, 2, 2882880, 30, 4620, 30, 210, 6, 120120, 210, 60060, 2310, 30, 6, 232792560, 6, 30, 2310, 30030, 30, 9240, 30, 2310, 2310, 510510, 6, 1396755360, 6, 210, 4620, 2310, 210, 120120, 6
OFFSET
1,3
COMMENTS
This sequence can be used as a filter for certain sequences involving Fibonacci numbers as it matches to any sequence that is obtained as f(A000045(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
FORMULA
a(n) = A046523(A000045(n)).
EXAMPLE
From Michael De Vlieger, May 18 2017: (Start)
a(6) = 8 because Fibonacci(6) = 8, the multiplicity of the prime factor of 8 is 3; the smallest p^3 = 2^3 = 8.
a(7) = 2 because Fibonacci(7) = 13, the multiplicity of the prime factor of 13 is 1; the smallest p^1 = 2^1 = 2.
a(15) = 30 because Fibonacci(15) = 610. The multiplicities of the prime factors of 610, in order from greatest to least, are {1, 1, 1}, the smallest prime power product p^1 * q^1 * r^1 = 2 * 3 * 5 = 30.
a(18) = 120 because Fibonacci(18) = 2584 = 2^3 * 17 * 19 -> 2^3 * 3 * 5 = 120. (End)
MATHEMATICA
Table[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &,
Sort[FactorInteger[#][[All, -1]], Greater]]] &@ Fibonacci@ n, {n, 79}] (* Michael De Vlieger, May 18 2017 *)
PROG
(PARI)
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
f0 = 0; f1 = 1; for(n=1, 10000, write("b278245.txt", n, " ", A046523(f1)); old_f0 = f0; f0 = f1; f1 = f1 + old_f0; );
(Scheme) (define (A278245 n) (A046523 (A000045 n)))
CROSSREFS
Cf. A286545 (rgs-version of this sequence), A286467.
Cf. A001605 (positions of 2's), A072381 (of 6's).
Sequences with matching equivalence classes: A063375, A105307, A152774.
Sequence in context: A029605 A367189 A054271 * A321026 A240284 A344897
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 16 2016
STATUS
approved