A028928 -id:A028928 - cp's OEIS frontend

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

Antti Karttunen, Nov 16 2016

nonn

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.

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.

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 *)

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

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

A106915 Primes of the form 3x^2 + 2xy + 5y^2, with x and y any integer.

Original entry on oeis.org

3, 5, 13, 19, 59, 61, 83, 101, 131, 139, 157, 173, 181, 227, 229, 251, 269, 283, 293, 307, 349, 397, 419, 461, 467, 509, 523, 563, 587, 619, 643, 661, 677, 691, 733, 773, 787, 797, 811, 829, 853, 859, 941, 971, 997, 1013, 1021, 1069, 1091, 1109, 1123

Offset: 1

T. D. Noe, May 09 2005

Discriminant = -56.
Also primes congruent to {3,5,13,19,27,45} mod 56. - Vincenzo Librandi, Jul 02 2016
The theta series for the quadratic form 3x^2 + 2xy + 5y^2 is the g.f. of A028928. - Michael Somos, Jul 02 2016
Legendre symbol (-14, a(n)) = Kronecker symbol (a(n), 14) = 1. Also, this sequence lists primes p such that Kronecker symbol (p, 2) = Legendre symbol (p, 7) = -1, i.e., primes p == 3, 5 (mod 8) and 3, 5, 6 (mod 7). - Jianing Song, Sep 04 2018

			59 is in the sequence since it is prime, and 59 = 3x^2 + 2xy + 5y^2 with x = 3 and y = 2. - _Michael B. Porter_, Jul 02 2016

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A028928.

[p: p in PrimesUpTo(3000) | p mod 56 in {3,5,13,19,27,45}]; // Vincenzo Librandi, Jul 02 2016

Union[QuadPrimes2[3, 2, 5, 10000], QuadPrimes2[3, -2, 5, 10000]] (* see A106856 *)
Select[Prime@Range[600], MemberQ[{3, 5, 13, 19, 27, 45}, Mod[#, 56]] &] (* Vincenzo Librandi, Jul 02 2016 *)

cp's OEIS Frontend

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

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