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%I #28 Sep 09 2017 19:23:03
%S 1,2,2,6,2,12,6,30,2,60,12,120,6,180,30,210,2,420,60,1080,12,2160,120,
%T 2520,6,2520,180,7560,30,6300,210,2310,2,4620,420,37800,60,90720,1080,
%U 75600,12,226800,2160,544320,120,453600,2520,138600,6,138600,2520,756000,180,2268000,7560,831600,30,415800,6300,2079000,210,485100,2310,30030,2,60060,4620
%N Filter-sequence for Stern polynomials: Least number with the same prime signature as A260443(n).
%C This sequence can be used for filtering certain Stern polynomial (see A125184, A260443) related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A260443(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under 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.
%C Some of these are listed on the last line ("Sequences that partition N into ...") of Crossrefs section.
%H Antti Karttunen, <a href="/A278243/b278243.txt">Table of n, a(n) for n = 0..2048</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%H <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F a(n) = A046523(A260443(n)).
%t a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]] - Boole[# == 1] &@ a@ n, {n, 0, 66}] (* _Michael De Vlieger_, May 12 2017 *)
%o (Scheme) (define (A278243 n) (A046523 (A260443 n)))
%Y Cf. A046523, A260443.
%Y Cf. also A278222, A278226, A278234, A278235, A278236, A278261.
%Y Sequences that partition or seem to partition N into same or coarser equivalence classes: A002487, A126606, A277314, A277315, A277325, A277326, A277700, A277705.
%Y The following are less certain: A007302 (not proved, but the first 10000 terms match), A072453, A110955 (uncertain, but related to A007302), A218799, A218800.
%Y Note that the base-2 related sequences A069010 and A277561 (= 2^A069010(n)) do not match, although at first it seems so, up to for at least 139 initial terms. Also A028928 belongs to a different family.
%K nonn
%O 0,2
%A _Antti Karttunen_, Nov 16 2016
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