A286034 Compound filter: a(n) = P(A046523(n), A161942(n)), where P(n,k) is sequence A000027 used as a pairing function.

1, 8, 3, 49, 8, 34, 3, 239, 124, 97, 8, 165, 30, 34, 34, 1051, 47, 1237, 17, 508, 21, 97, 8, 727, 565, 331, 74, 165, 122, 733, 3, 4403, 34, 502, 34, 7911, 192, 196, 72, 2302, 233, 526, 68, 508, 1237, 97, 8, 3051, 1774, 5368, 97, 1782, 380, 727, 97, 727, 51, 1231, 122, 3220, 498, 34, 288, 18019, 331, 733, 155, 2713, 34, 733, 47, 35317, 705, 1897, 873, 1047, 34

Offset: 1

Author

Antti Karttunen, May 07 2017

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• MathWorld, Pairing Function

Crossrefs

Cf. A000027, A046523, A161942, A286260.

Programs

PARI

A000265(n) = (n >> 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
A161942(n) = A000265(sigma(n));
A286034(n) = (2 + ((A046523(n)+A161942(n))^2) - A046523(n) - 3*A161942(n))/2;
for(n=1, 16384, write("b286034.txt", n, " ", A286034(n)));

Python

from sympy import factorint, divisors, divisor_sigma
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 a000265(n): return max(list(filter(lambda i: i%2 == 1, divisors(n))))
def a161942(n): return a000265(divisor_sigma(n))
def a(n): return T(a046523(n), a161942(n)) # Indranil Ghosh, May 07 2017

Scheme

(define (A286034 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A161942 n)) 2) (- (A046523 n)) (- (* 3 (A161942 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A046523(n)+A161942(n))^2) - A046523(n) - 3*A161942(n)).