A278245 Least number with the same prime signature as the n-th Fibonacci number: a(n) = A046523(A000045(n)).

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.

Links

Antti Karttunen (terms 1..374) & Hans Havermann, Table of n, a(n) for n = 1..1300

Index entries for sequences computed from exponents in factorization of n

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. A000045, A046523, A278241, A278248.

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

Adjacent sequences: A278242 A278243 A278244 * A278246 A278247 A278248

Keyword

nonn

Author

Antti Karttunen, Nov 16 2016

Status

approved