A286369 - OEIS

%I #11 May 09 2017 17:36:35

%S 2,2,4,2,7,5,5,2,14,6,4,4,7,5,11,2,6,14,4,7,33,5,5,5,20,6,58,5,7,11,5,

%T 2,32,6,10,14,7,5,11,6,6,32,4,4,25,5,5,4,14,20,10,7,7,59,11,5,32,6,4,

%U 11,7,5,135,2,42,32,4,6,33,11,5,14,6,6,28,4,33,11,5,7,242,6,4,33,43,5,11,5,6,24,10,5,33,5,11,5,6,14,134,20,7,11,5,6,46,6

%N Compound filter: a(n) = 2*A286364(n) + floor(A072400(n)/4).

%C This sequence contains, in addition to the information contained in A286364 (which packs the values of A286361(n) and A286363(n) to a single value with the pairing function A000027) also the bit-2 of A072400(n) (its third least significant bit), which is here stored as the least significant bit of a(n). In contrast to A286366, the parity of the highest power of 2 dividing n is not stored.

%C Thus we have (among other such identities) the following two identities related to equivalence class partitioning:

%C For all odd i, odd j: a(i) = a(j) <=> A286366(i) = A286366(j).

%C For all odd i, odd j: a(i) = a(j) => A010877(i) = A010877(j). [On odd numbers the information contained in a(n) is sufficient to determine the value of n modulo 8, one of the 1, 3, 5 or 7.]

%H Antti Karttunen, <a href="/A286369/b286369.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 2*A286364(n) + floor(A072400(n)/4).

%o (Scheme) (define (A286369 n) (+ (* 2 (A286364 n)) (floor->exact (/ (A072400 n) 4))))

%o (Python)

%o from sympy.ntheory.factor_ import digits

%o from sympy import factorint

%o from operator import mul

%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, k):

%o     f = factorint(n)

%o     return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])

%o def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2

%o def a286364(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3)))

%o def a072400(n): return int(str(int(''.join(map(str, digits(n, 4)[1:]))[::-1]))[::-1], 4)%8

%o def a(n): return 2*a286364(n) + int(a072400(n)/4) # _Indranil Ghosh_, May 09 2017

%Y Cf. A072400, A286364, A286366, A286370, A286371.

%K nonn

%O 1,1

%A _Antti Karttunen_, May 09 2017