A278245 - OEIS

%I #37 May 19 2017 06:02:53

%S 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,

%T 210,210,2,9240,6,210,30,6,30,166320,30,30,30,30030,6,9240,2,2310,

%U 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

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

%C 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 ...").

%C 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.

%H Antti Karttunen (terms 1..374) & Hans Havermann, <a href="/A278245/b278245.txt">Table of n, a(n) for n = 1..1300</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = A046523(A000045(n)).

%e From _Michael De Vlieger_, May 18 2017: (Start)

%e a(6) = 8 because Fibonacci(6) = 8, the multiplicity of the prime factor of 8 is 3; the smallest p^3 = 2^3 = 8.

%e a(7) = 2 because Fibonacci(7) = 13, the multiplicity of the prime factor of 13 is 1; the smallest p^1 = 2^1 = 2.

%e 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.

%e a(18) = 120 because Fibonacci(18) = 2584 = 2^3 * 17 * 19 -> 2^3 * 3 * 5 = 120. (End)

%t Table[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &,

%t Sort[FactorInteger[#][[All, -1]], Greater]]] &@ Fibonacci@ n, {n, 79}] (* _Michael De Vlieger_, May 18 2017 *)

%o (PARI)

%o 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

%o f0 = 0; f1 = 1; for(n=1, 10000, write("b278245.txt", n, " ", A046523(f1)); old_f0 = f0; f0 = f1; f1 = f1 + old_f0; );

%o (Scheme) (define (A278245 n) (A046523 (A000045 n)))

%Y Cf. A000045, A046523, A278241, A278248.

%Y Cf. A286545 (rgs-version of this sequence), A286467.

%Y Cf. A001605 (positions of 2's), A072381 (of 6's).

%Y Sequences with matching equivalence classes: A063375, A105307, A152774.

%K nonn

%O 1,3

%A _Antti Karttunen_, Nov 16 2016