A286252 Compound filter: a(n) = P(A001511(1+n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 5, 2, 18, 2, 23, 7, 59, 2, 23, 16, 94, 7, 80, 29, 195, 2, 23, 16, 94, 16, 467, 67, 355, 7, 80, 67, 706, 29, 302, 121, 672, 2, 23, 16, 94, 16, 467, 67, 355, 16, 467, 436, 1894, 67, 1832, 277, 1331, 7, 80, 67, 706, 67, 1832, 631, 2779, 29, 302, 277, 2704, 121, 1178, 497, 2422, 2, 23, 16, 94, 16, 467, 67, 355, 16, 467, 436, 1894, 67, 1832, 277, 1331, 16, 467, 436

Offset: 0

Author

Antti Karttunen, May 07 2017

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16384
• MathWorld, Pairing Function

Crossrefs

Cf. A000027, A001511, A278222, A286162, A286251, A286253, A286254.

Programs

PARI

A001511(n) = (1+valuation(n,2));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A286252(n) = (2 + ((A001511(1+n)+A278222(n))^2) - A001511(1+n) - 3*A278222(n))/2;
for(n=0, 16384, write("b286252.txt", n, " ", A286252(n)));

Python

from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
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 a278222(n): return a046523(a005940(n + 1))
def a001511(n): return 2 + bin(n - 1)[2:].count("1") - bin(n)[2:].count("1")
def a(n): return T(a001511(n + 1), a278222(n)) # Indranil Ghosh, May 07 2017

Scheme

(define (A286252 n) (* (/ 1 2) (+ (expt (+ (A001511 (+ 1 n)) (A278222 n)) 2) (- (A001511 (+ 1 n))) (- (* 3 (A278222 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A001511(1+n)+A278222(n))^2) - A001511(1+n) - 3*A278222(n)).