A286367 Compound filter: a(n) = P(A001511(n), A286364(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 3, 2, 6, 4, 5, 2, 10, 22, 8, 2, 9, 4, 5, 11, 15, 4, 30, 2, 13, 121, 5, 2, 14, 46, 8, 407, 9, 4, 17, 2, 21, 121, 8, 11, 39, 4, 5, 11, 19, 4, 138, 2, 9, 67, 5, 2, 20, 22, 57, 11, 13, 4, 437, 11, 14, 121, 8, 2, 24, 4, 5, 2212, 28, 211, 138, 2, 13, 121, 17, 2, 49, 4, 8, 92, 9, 121, 17, 2, 26, 7261, 8, 2, 156, 211, 5, 11, 14, 4, 80, 11, 9, 121, 5, 11, 27, 4, 30

Offset

1,2

Comments

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 highest power of 2 dividing n. Note that this is more information than A286365, as it stores only the parity of the exponent of 2.

For all i, j: a(i) = a(j) => A286161(i) = A286161(j).

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pairing Function

Formula

a(n) = (1/2)*(2 + ((A001511(n)+A286364(n))^2) - A001511(n) - 3*A286364(n)).

Prog

(Scheme) (define (A286367 n) (* (/ 1 2) (+ (expt (+ (A001511 n) (A286364 n)) 2) (- (A001511 n)) (- (* 3 (A286364 n))) 2)))
(Python)
from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def A(n, k):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==k else i**f[i] for i in f])
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a286364(n): return T(a046523(n/A(n, 1)), a046523(n/A(n, 3)))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n), a286364(n)) # Indranil Ghosh, May 09 2017

Crossrefs

Cf. A000027, A001511, A286161, A286361, A286363, A286364, A286365.

Sequence in context: A360259 A340873 A033940 * A196047 A106409 A115510

Adjacent sequences: A286364 A286365 A286366 * A286368 A286369 A286370

Keyword

nonn

Author

Antti Karttunen, May 08 2017

Status

approved