A323275 Let f(p, q) denote the pair (p + q, wt(p) + wt(q)); a(n) is obtained by iterating f starting at (n, 1) until p/q is an integer (and then a(n) is that integer), or if no integer is ever reached then a(n) = -1. (Here wt is binary weight, A000120.)

1, 2, 2, 2, 2, 2, 2, 10, 7, 10, 3, 7, 5, 10, 7, 22, 6, 22, 5, 7, 8, 22, 10, 10, 8, 8, 10, 10, 6, 8, 22, 8, 22, 8, 9, 8, 10, 8, 8, 22, 10, 22, 22, 22, 22, 22, 8, 15, 22, 11, 15, 15, 22, 11, 16, 16, 22, 15, 10, 16, 15, 22, 15, 14, 22, 14, 17, 23, 40, 15, 22, 22, 40, 12, 22, 22, 16, 12, 18, 27, 18, 40, 40, 40, 22, 40, 40, 14, 18, 34

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Jeffrey C. Lagarias, Wild and Wooley numbers, The American Mathematical Monthly, Vol. 113, No. 2 (2006), pp. 97-108; arXiv preprint, arXiv:math/0411141 [math.NT], 2004-2005.

Example

(8, 1) -> (9, 2) -> (11, 3) -> (14, 5) -> (19, 5) -> (24, 5) -> (29, 4) -> (33, 5) -> (38, 4) -> (42, 4) -> (46, 4) -> (50, 5). 50/5 is an integer, so a(8) = 50/5 = 10.

Mathematica

a[n_] := Divide@@ NestWhile[{Total[#], Total[DigitCount[#, 2, 1]]}&, {n, 1}, Last[#] == 1 || !Divisible@@# &]; Array[a, 100] (* Amiram Eldar, Jul 29 2025 *)

Prog

(PARI) f(v) = return([v[1]+v[2], hammingweight(v[1])+hammingweight(v[2])]);
a(n) = {my(nb = 0, v = [n, 1]); while (1, v = f(v); nb++; if (frac(q=v[1]/v[2]) == 0, return (q))); } \\ Michel Marcus, Jan 13 2019
(Python)
from itertools import count
def f(p, q): return (p+q, p.bit_count() + q.bit_count())
def a(n):
    t = n, 1
    return next(t[0]//t[1] for i in count(1) if (t:=f(*t))[0]%t[1] == 0)
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Dec 23 2025

Crossrefs

Cf. A000120, A059175, A323375.

Sequence in context: A339165 A129381 A319991 * A139514 A377352 A330955

Adjacent sequences: A323272 A323273 A323274 * A323276 A323277 A323278

Keyword

nonn,base

Author

Ctibor O. Zizka, Jan 12 2019

Extensions

Missing term a(87) inserted by Amiram Eldar, Jul 29 2025

Status

approved