A286362 -id:A286362 - cp's OEIS frontend

A286365 Compound filter: a(n) = 2*A286364(n) + A000035(A007814(n)).

2, 3, 4, 2, 6, 5, 4, 3, 14, 7, 4, 4, 6, 5, 10, 2, 6, 15, 4, 6, 32, 5, 4, 5, 20, 7, 58, 4, 6, 11, 4, 3, 32, 7, 10, 14, 6, 5, 10, 7, 6, 33, 4, 4, 24, 5, 4, 4, 14, 21, 10, 6, 6, 59, 10, 5, 32, 7, 4, 10, 6, 5, 134, 2, 42, 33, 4, 6, 32, 11, 4, 15, 6, 7, 28, 4, 32, 11, 4, 6, 242, 7, 4, 32, 42, 5, 10, 5, 6, 25, 10, 4, 32, 5, 10, 5, 6, 15, 134, 20, 6, 11, 4, 7, 46, 7

Offset: 1

Antti Karttunen, May 08 2017

nonn

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 information whether the exponent of the highest power of 2 dividing n is even or odd, which is stored in the least significant bit of a(n). Thus, for example, all squares (A000290) can be obtained by listing such numbers n that a(n) is even and both A002260(a(n)/2) & A004736(a(n)/2) are perfect squares.

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

Cf. A000035, A007814, A035263, A286361, A286362, A286363, A286364, A286369.

Cf. A286366, A286367 (similar, but contain more information).

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 a007814(n): return 1 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return 2*a286364(n) + a007814(n)%2 # Indranil Ghosh, May 09 2017

(define (A286365 n) (+ (* 2 (A286364 n)) (A000035 (A007814 n))))

a(n) = (2*A286364(n)) + (1 - A035263(n)) = 2*A286364(n) + A000035(A007814(n)).

A286462 Compound filter (3-adic valuation & the length of rightmost run of 1's in base-2): a(n) = P(A051064(n), A089309(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 1, 5, 1, 1, 5, 4, 1, 6, 1, 2, 5, 1, 4, 12, 1, 1, 6, 2, 1, 3, 2, 4, 5, 1, 1, 14, 4, 1, 12, 11, 1, 3, 1, 2, 6, 1, 2, 8, 1, 1, 3, 2, 2, 6, 4, 7, 5, 1, 1, 5, 1, 1, 14, 4, 4, 3, 1, 2, 12, 1, 11, 31, 1, 1, 3, 2, 1, 3, 2, 4, 6, 1, 1, 5, 2, 1, 8, 7, 1, 15, 1, 2, 3, 1, 2, 8, 2, 1, 6, 2, 4, 3, 7, 11, 5, 1, 1, 9, 1, 1, 5, 4, 1, 3, 1, 2, 14, 1, 4, 12, 4, 1, 3, 2, 1

Offset: 1

Antti Karttunen, May 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A051064, A089309, A286362, A286463, A286464.

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A089309(n) = valuation((n/2^valuation(n, 2))+1, 2); \\ After Ralf Stephan
A286462(n) = (1/2)*(2 + ((A051064(n)+A089309(n))^2) - A051064(n) - 3*A089309(n));
for(n=1, 10000, write("b286462.txt", n, " ", A286462(n)));

from sympy import divisors, divisor_count, mobius
def a051064(n): return -sum([mobius(3*d)*divisor_count(n/d) for d in divisors(n)])
def v(n): return bin(n)[2:][::-1].index("1")
def a089309(n): return  0 if n==0 else v(n/2**v(n) + 1)
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a(n): return T(a051064(n), a089309(n)) # Indranil Ghosh, May 11 2017

(define (A286462 n) (* (/ 1 2) (+ (expt (+ (A051064 n) (A089309 n)) 2) (- (A051064 n)) (- (* 3 (A089309 n))) 2)))

a(n) = (1/2)*(2 + ((A051064(n)+A089309(n))^2) - A051064(n) - 3*A089309(n)).

cp's OEIS Frontend

A286365 Compound filter: a(n) = 2*A286364(n) + A000035(A007814(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula