A278243 Filter-sequence for Stern polynomials: Least number with the same prime signature as A260443(n).

1, 2, 2, 6, 2, 12, 6, 30, 2, 60, 12, 120, 6, 180, 30, 210, 2, 420, 60, 1080, 12, 2160, 120, 2520, 6, 2520, 180, 7560, 30, 6300, 210, 2310, 2, 4620, 420, 37800, 60, 90720, 1080, 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

Offset: 0

Author

Antti Karttunen, Nov 16 2016

Keywords

• nonn

Comments

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 ...").

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.

Some of these are listed on the last line ("Sequences that partition N into ...") of Crossrefs section.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..2048
• Index entries for sequences computed from exponents in factorization of n
• Index entries for sequences related to Stern's sequences

Crossrefs

Cf. A046523, A260443.

Cf. also A278222, A278226, A278234, A278235, A278236, A278261.

Sequences that partition or seem to partition N into same or coarser equivalence classes: A002487, A126606, A277314, A277315, A277325, A277326, A277700, A277705.

The following are less certain: A007302 (not proved, but the first 10000 terms match), A072453, A110955 (uncertain, but related to A007302), A218799, A218800.

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.

Programs

Mathematica

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 *)

Scheme

(define (A278243 n) (A046523 (A260443 n)))

Formula

a(n) = A046523(A260443(n)).