A059175 -id:A059175 - cp's OEIS frontend

A214866 Numbers m such that A059175(m) = 0.

0, 84, 102, 141, 167, 173, 228, 246, 250, 358, 366, 372, 388, 421, 424, 444, 502, 599, 610, 617, 640, 653, 660, 685, 804, 865, 866, 867, 875, 920, 941, 1002, 1041, 1059, 1067, 1162, 1186, 1208, 1236, 1238, 1308, 1409, 1445, 1464, 1473, 1523, 1640, 1671, 1757

Offset: 1

Reinhard Zumkeller, Mar 11 2013

nonn

A059175(a(n)) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A216183.

import Data.List (elemIndices)
a214866 n = a214866_list !! (n-1)
a214866_list = elemIndices 0 a059175_list

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.)

Original entry on oeis.org

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

A059514 For a rational number p/q let f(p/q) = p*q divided by (the sum of digits of p and of q) minus 1; a(n) is obtained by iterating f, starting at n/1, until an integer is reached, or if no integer is ever reached then a(n) = 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 4, 28, 42, 7315, 208, 136, 2, 19, 10, 7, 11, 69, 4, 2310, 28, 3, 42, 319, 10, 189885850, 96, 11, 323323, 205530, 4, 37, 228, 28, 10, 123, 7, 559, 11, 5, 69, 517, 4, 152152, 10, 187, 28, 424, 6, 11, 154, 0, 77140, 2478, 10, 0

Offset: 1

Floor van Lamoen, Jan 22 2001

a(A216183(n)) = 0. - Reinhard Zumkeller, Mar 11 2013

			14/1 -> 14/5 -> 70/9 -> 630/15 = 42 so a(14)=42.
57/1 -> 19/4 -> 76/13 -> 247/4 -> 247/4 -> ...  so a(57) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A059175, A058971, A058972, A058977, A058988.

Cf. A007953.

import Data.Ratio ((%), numerator, denominator)
a059514 n = f [n % 1] where
   f xs@(x:_)
     | denominator y == 1 = numerator y
     | y `elem` xs        = 0
     | otherwise          = f (y : xs)
     where y = (numerator x * denominator x) %
               (a007953 (numerator x) + a007953 (denominator x) - 1)
-- Reinhard Zumkeller, Mar 11 2013

Corrected and extended by Naohiro Nomoto, Jul 20 2001

A145833 The "Wild Numbers", from the novel of the same title (Version 2).

Original entry on oeis.org

11, 67, 4769, 67

Offset: 1

Aaron Swartz (me(AT)aaronsw.com), Oct 20 2008

nonn

Apparently these are completely fictional and there is no mathematical explanation. However, see the pseudo-wild numbers in A058971, A058972, A058973, A058977, A058988, A059175. See also the Lagarias article.

P. Schogt, De Wilde Getallen, De Arbeiderspers, Amsterdam, 1998.
P. Schogt, The Wild Numbers, Four Walls Eight Windows Pub., New York, 2000.
D. F. Wallace, Rhetoric and the math melodrama, Science, 290 (Dec 22 2000), 2263-2267.
A number of other reviews of this book exist on the Web.

J. C. Lagarias, Wild and Wooley numbers, Amer. Math. Monthly, 113 (No. 2, 2006), 97-108.
Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.
P. Schogt, The Wild Number Problem: math or fiction?, arXiv preprint arXiv:1211.6583 [math.HO], 2012. - From _N. J. A. Sloane_, Jan 03 2013
Another review

A variant of A058883, which is the main entry for this sequence.

A058883 The "Wild Numbers", from the novel of the same title (Version 1).

Original entry on oeis.org

11, 67, 2, 4769, 67

Offset: 0

N. J. A. Sloane, Jan 08 2001

nonn

Apparently these are completely fictional and there is no mathematical explanation. However, see the pseudo-wild numbers in A058971, A058972, A058973, A058977, A058988, A059175. See also the Lagarias article.

P. Schogt, De Wilde Getallen, De Arbeiderspers, Amsterdam, 1998.
P. Schogt, The Wild Numbers, Four Walls Eight Windows Pub., New York, 2000.
D. F. Wallace, Rhetoric and the math melodrama, Science, 290 (Dec 22 2000), 2263-2267.
A number of other reviews of this book exist on the Web.

J. C. Lagarias, Wild and Wooley numbers, Amer. Math. Monthly, 113 (No. 2, 2006), 97-108.
Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.
P. Schogt, The Wild Number Problem: math or fiction?, arXiv:1211.6583 [math.HO], 2012. - From _N. J. A. Sloane_, Jan 03 2013
Wikipedia, Wild number
Another review

See A145833 for another version.

Thanks to Enoch Haga for investigating these numbers (Jan 14 2001).

Offset changed to 0 by Sean A. Irvine, Sep 03 2022, because of the Wikipedia link.

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 *)

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A214866 Numbers m such that A059175(m) = 0.

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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.)

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A059514 For a rational number p/q let f(p/q) = p*q divided by (the sum of digits of p and of q) minus 1; a(n) is obtained by iterating f, starting at n/1, until an integer is reached, or if no integer is ever reached then a(n) = 0.

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A145833 The "Wild Numbers", from the novel of the same title (Version 2).

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A058883 The "Wild Numbers", from the novel of the same title (Version 1).

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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.

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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)).

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