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

Author

Ctibor O. Zizka, Jan 12 2019

Keywords

• nonn
• base

Examples

			(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.
		

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.

Crossrefs

Cf. A000120, A059175, A323375.

Programs

Mathematica

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

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

Extensions

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