A286240 Compound filter: a(n) = P(A278222(n), A278222(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

2, 5, 12, 14, 23, 42, 59, 44, 23, 61, 142, 117, 109, 183, 261, 152, 23, 61, 142, 148, 601, 850, 607, 375, 109, 265, 1093, 939, 473, 765, 1097, 560, 23, 61, 142, 148, 601, 850, 607, 430, 601, 1741, 3946, 2545, 2497, 3463, 2509, 1323, 109, 265, 1093, 1117, 2497, 4525, 5707, 3153, 473, 1105, 4489, 3813, 1969, 3129, 4497, 2144, 23, 61, 142, 148, 601, 850, 607, 430, 601, 1741

Offset

0,1

Links

Antti Karttunen, Table of n, a(n) for n = 0..16383

Eric Weisstein's World of Mathematics, Pairing Function

Formula

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

Prog

(PARI)
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));
A286240(n) = (2 + ((A278222(n)+A278222(1+n))^2) - A278222(n) - 3*A278222(1+n))/2;
for(n=0, 16383, write("b286240.txt", n, " ", A286240(n)));
(Scheme) (define (A286240 n) (* (/ 1 2) (+ (expt (+ (A278222 n) (A278222 (+ 1 n))) 2) (- (A278222 n)) (- (* 3 (A278222 (+ 1 n)))) 2)))
(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 a(n): return T(a278222(n), a278222(n + 1)) # Indranil Ghosh, May 07 2017

Crossrefs

Cf. A000027, A278222, A286241, A286242, A286255.

Cf. A088705 (one of the matches not matched by A278222 alone. Thus also the whole A007814 (A001511) family is included).

Sequence in context: A286255 A286160 A286163 * A215974 A192524 A287553

Adjacent sequences: A286237 A286238 A286239 * A286241 A286242 A286243

Keyword

nonn

Author

Antti Karttunen, May 07 2017

Status

approved