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%I #33 Sep 23 2023 12:11:52
%S 1,2,2,2,4,2,2,6,2,2,6,2,4,8,2,2,6,6,2,6,2,2,12,2,4,6,2,6,6,2,2,12,6,
%T 2,6,2,2,12,6,2,16,2,6,6,2,6,6,6,2,12,2,2,30,2,2,6,2,6,12,6,4,6,8,2,6,
%U 2,6,24,2,2,6,6,6,12,2,2,12,6,2,6,6,2,30,2,4,12,2,12,6,2,2,6,6,6,24,2,2,30,2,2,6,6,6,12,6,2,6,6,6,6,6,2,36,2,2
%N Least number with the same prime signature as the n-th odd number: a(n) = A046523(2n-1).
%C This sequence works as a filter for sequences related to the prime factorization of odd numbers by matching to any sequence that is obtained as f(2*n - 1), 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 ..."). The last line in Crossrefs section lists such sequences that were present in the database as of Nov 11 2016, although some of the matches might be spurious.
%H Antti Karttunen, <a href="/A278223/b278223.txt">Table of n, a(n) for n = 1..32769</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F a(n) = A046523(2n - 1).
%F a(n) = A046523(A064216(n)).
%F From _Antti Karttunen_, May 31 2017: (Start)
%F a(n) = A278222(A244153(n)).
%F a(n) = A278531(A245611(n)).
%F (End)
%t a[n_] := Times @@ (Prime[Range[Length[f = FactorInteger[2*n - 1]]]]^Sort[f[[;; , 2]], Greater]); a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Jul 23 2023 *)
%o (Scheme)
%o (define (A278223 n) (A046523 (+ n n -1)))
%o (define (A278223 n) (A046523 (A064216 n)))
%o (Python)
%o from sympy import factorint
%o def P(n):
%o f = factorint(n)
%o return sorted([f[i] for i in f])
%o def a046523(n):
%o x=1
%o while True:
%o if P(n) == P(x): return x
%o else: x+=1
%o def a(n): return a046523(2*n - 1) # _Indranil Ghosh_, May 11 2017
%o (Python)
%o from math import prod
%o from sympy import prime, factorint
%o def A278223(n): return prod(prime(i+1)**e for i,e in enumerate(sorted(factorint((n<<1)-1).values(),reverse=True))) # _Chai Wah Wu_, Sep 16 2022
%Y Odd bisection of A046523.
%Y Cf. A064216, A244153, A245611, A278222, A278224, A278531.
%Y Sequences that partition or seem to partition N into same or coarser equivalence classes: A099774, A100007, A193773, A101871, A158280, A158315, A158647, A285716.
%K nonn
%O 1,2
%A _Antti Karttunen_, Nov 16 2016
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