A323375 -id:A323375 - cp's OEIS frontend

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

Ctibor O. Zizka, Jan 12 2019

			(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.

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.

Cf. A000120, A059175, A323375.

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

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

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

A323356 For a rational number p/q let f(p/q) = (p+q) / (A000120(p) + A000120(q)); a(n) is obtained by iterating f, starting at n/1, until an integer is reached (and then a(n) = that integer), or if no integer is ever reached then a(n) = -1.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, -1, 7, -1, 3, 7, 7, -1, 7, -1, 6, -1, 5, 7, 7, -1, -1, -1, -1, 8, -1, -1, 6, 8, -1, 8, 11, 8, 9, 8, -1, 8, 8, 11, -1, 11, 11, 11, -1, 11, 8, -1, 8, 11, -1, -1, 11, 11, 8, 16, -1, 15, 10, 16, -1, -1, 15, 14, 22, 14, 17, 23, 11, 15, 11, 11, 8, 12, 11, 11, 16, 12, 11

Offset: 1

Ctibor O. Zizka, Jan 18 2019

			13/1 -> 14/4=7/2 -> 9/4 -> 13/3 -> 16/5 -> 21/3 = 7 so a(13) = 7.
8/1 -> 9/2 -> 11/3 -> 14/5 -> 19/5 -> 24/5 -> 29/4 -> 33/5 -> 38/4=19/2 -> 21/4 -> 25/4 -> 29/4 and the 5-cycle repeats, so a(8) = -1.

Cf. A000120, A058971, A059175, A323275, A323375.

Array[SelectFirst[Rest@ NestWhileList[(#1 + #2)/(DigitCount[#1, 2, 1] + DigitCount[#2, 2, 1]) & @@ {Numerator@ #, Denominator@ #} &, #, UnsameQ, All], IntegerQ] /. k_ /; MissingQ@ k -> -1 &, 79] (* Michael De Vlieger, Jan 18 2019 *)

A323596 Number of (positive) iterations of f to reach an integer when starting from n/1. If no integer is ever reached then a(n) = -1. f(p/q) = (p + q) / (A000120(p) + A000120(q)).

Original entry on oeis.org

1, 3, 3, 3, 1, 2, 1, -1, 4, -1, 1, 3, 5, -1, 2, -1, 1, -1, 1, 1, 4, -1, -1, -1, -1, 5, -1, -1, 1, 4, -1, 5, 16, 4, 1, 2, -1, 3, 1, 14, -1, 13, 13, 13, -1, 12, 1, -1, 6, 2, -1, -1, 11, 1, 5, 13, -1, 4, 1, 12, -1, -1, 3, 3, 1, 2, 1, 1, 16, 2, 8, 8, 4, 3, 7, 7, 9, 2, 14

Offset: 1

Ctibor O. Zizka, Jan 18 2019

sign

			8/1 -> 9/2 -> 11/3 -> 14/5 -> 19/5 -> 24/5 -> 29/4 -> 33/5 -> 38/4=19/2 -> 21/4 -> 25/4 -> 29/4 and the 5-cycle repeats, so a(8) = -1.
13/1 -> 14/4=7/2 -> 9/4 -> 13/3 -> 16/5 -> 21/3=7 so a(13) = 5.

Cf. A000120, A058971, A059175, A323275, A323356, A323375.

Array[If[AnyTrue[#, IntegerQ], 1 + LengthWhile[#, ! IntegerQ@ # &], -1] &@ Rest@ NestWhileList[(#1 + #2)/(DigitCount[#1, 2, 1] + DigitCount[#2, 2, 1]) & @@ {Numerator@ #, Denominator@ #} &, #, UnsameQ, All] &, 79] (* Michael De Vlieger, Jan 18 2019 *)

cp's OEIS Frontend

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.)

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A323356 For a rational number p/q let f(p/q) = (p+q) / (A000120(p) + A000120(q)); a(n) is obtained by iterating f, starting at n/1, until an integer is reached (and then a(n) = that integer), or if no integer is ever reached then a(n) = -1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A323596 Number of (positive) iterations of f to reach an integer when starting from n/1. If no integer is ever reached then a(n) = -1. f(p/q) = (p + q) / (A000120(p) + A000120(q)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica