A286379 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Eric Weisstein's World of Mathematics, Pairing Function

FORMULA

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

PROG

(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)));

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

(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):