A286164 Compound filter: a(n) = T(A055396(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

0, 2, 5, 7, 9, 16, 14, 29, 12, 16, 20, 67, 27, 16, 23, 121, 35, 67, 44, 67, 23, 16, 54, 277, 18, 16, 38, 67, 65, 436, 77, 497, 23, 16, 31, 631, 90, 16, 23, 277, 104, 436, 119, 67, 80, 16, 135, 1129, 25, 67, 23, 67, 152, 277, 31, 277, 23, 16, 170, 1771, 189, 16, 80, 2017, 31, 436, 209, 67, 23, 436, 230, 2557, 252, 16, 80, 67, 40, 436, 275, 1129, 138, 16, 299

Offset: 1

Author

Antti Karttunen, May 04 2017

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• MathWorld, Pairing Function

Crossrefs

Cf. A000027, A046523, A055396, A286160, A286161, A286162, A286163.

Programs

PARI

A001511(n) = (1+valuation(n,2));
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
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286164(n) = (2 + ((A055396(n)+A046523(n))^2) - A055396(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286164.txt", n, " ", A286164(n)));

Python

from sympy import primepi, isprime, primefactors, factorint
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
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): return T(a055396(n), a046523(n)) # Indranil Ghosh, May 05 2017

Scheme

(define (A286164 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A046523 n)) 2) (- (A055396 n)) (- (* 3 (A046523 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A055396(n)+A046523(n))^2) - A055396(n) - 3*A046523(n)).