A286379 Compound filter ("discard the smallest prime factor" & "signature for 1-runs in base-2"): a(n) = P(A032742(n), A278222(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = 1.

1, 2, 7, 5, 16, 18, 29, 14, 31, 50, 67, 42, 67, 98, 195, 44, 16, 100, 67, 115, 637, 242, 277, 117, 125, 289, 955, 224, 277, 450, 497, 152, 131, 248, 160, 271, 436, 454, 643, 320, 436, 1246, 1771, 550, 2716, 1058, 1129, 375, 160, 655, 1343, 692, 1771, 1918, 3332, 623, 880, 1355, 2557, 1020, 1129, 1922, 3507, 560, 166, 736, 67, 775, 1349, 1070, 277, 856, 436

Offset: 1

Author

Antti Karttunen, May 13 2017

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Eric Weisstein's World of Mathematics, Pairing Function

Crossrefs

Cf. A032742, A278222, A285729.

Programs

PARI

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
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));
A286379(n) = if(1==n,n,(1/2)*(2 + ((A032742(n)+A278222(n))^2) - A032742(n) - 3*A278222(n)));
for(n=1, 16384, write("b286379.txt", n, " ", A286379(n)));

Python

from sympy import factorint, divisors
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 a(n): return 1 if n==1 else T(divisors(n)[-2], a278222(n)) # Indranil Ghosh, May 13 2017

Scheme

(define (A286379 n) (if (= 1 n) n (* (/ 1 2) (+ (expt (+ (A032742 n) (A278222 n)) 2) (- (A032742 n)) (- (* 3 (A278222 n))) 2))))

Formula

a(1) = 1, for n > 1, a(n) = (1/2)*(2 + ((A032742(n)+A278222(n))^2) - A032742(n) - 3*A278222(n)).