A058971 -id:A058971 - cp's OEIS frontend

A058972 For a rational number p/q let f(p/q) = sum of aliquot divisors of p+q divided by number of divisors of p+q; sequence gives numbers k such that, starting at k/1 and iterating f, an integer is eventually reached.

3, 9, 15, 24, 25, 29, 33, 35, 50, 51, 55, 57, 59, 63, 73, 79, 80, 81, 85, 87, 89, 90, 95, 99, 105, 119, 120, 121, 128, 131, 139, 143, 145, 169, 177, 179, 181, 183, 193, 195, 201, 203, 204, 211, 215, 217, 218, 219, 221, 225, 227, 233, 247, 248, 255, 273, 275, 288

Offset: 1

N. J. A. Sloane, Jan 14 2001

			f(9/1) = 8/4 = 2, an integer, so 9 is in the sequence;
f(10/1) = 1/2 and f(1/2)=1/2, so 10 is not in the sequence.

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

Cf. A058971, A058973.

Cf. A001065, A000005.

import Data.Ratio ((%), numerator, denominator)
a058972 n = a058972_list !! (n-1)
a058972_list = map numerator $ filter ((f [])) [1..] where
   f ys q = denominator y == 1 || not (y `elem` ys) && f (y : ys) y
            where y = a001065 q' % a000005 q'
                  q' = numerator q + denominator q
-- Reinhard Zumkeller, Jun 15 2013

f[r_] := If[init == False && IntegerQ[r], r, init = False; p = Numerator[r]; q = Denominator[r]; d = Most[Divisors[p+q]]; Total[d]/(Length[d]+1)]; ok[n_] := IntegerQ[ init = True; FixedPoint[f, n/1]]; ok[1] = False; A058972 = Select[ Range[300], ok] (* Jean-François Alcover, Dec 21 2011 *)

f2(p,q) = (sigma(p+q)-p-q)/numdiv(p+q);
f1(r) = f2(numerator(r), denominator(r));
loop(list) = {my(v=Vecrev(list)); for (i=2, #v, if (v[i] == v[1], return(1)););}
ff(n) = {my(ok=0, m=f2(n,1), list=List()); while(denominator(m) != 1, m = f1(m); listput(list, m); if (loop(list), return (0));); return(m);}
isok(m) = ff(m) > 0; \\ Michel Marcus, Feb 09 2022

Corrected and extended by Matthew Conroy, Apr 18 2001

A058977 For a rational number p/q let f(p/q) = sum of distinct prime factors (A008472) of p+q divided by number of distinct prime factors (A001221) of p+q; 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

2, 3, 2, 5, 7, 7, 2, 3, 3, 11, 7, 13, 11, 4, 2, 17, 7, 19, 3, 5, 4, 23, 7, 5, 17, 3, 11, 29, 13, 31, 2, 7, 5, 6, 7, 37, 23, 8, 3, 41, 4, 43, 4, 4, 3, 47, 7, 7, 3, 10, 17, 53, 7, 8, 11, 11, 7, 59, 13, 61, 6, 5, 2, 9, 19, 67, 5, 13, 17, 71, 7, 73, 41, 4, 23, 9, 6, 79, 3, 3, 4, 83, 4, 11, 47

Offset: 1

N. J. A. Sloane, Jan 14 2001

A247462 gives number of iterations needed to reach a(n). - Reinhard Zumkeller, Sep 17 2014

			f(5/1) = 5/2 and f(5/2) = 7, so a(5)=7.

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

Cf. A058971, A058972.
Cf. A008472, A001221.
Cf. A247462, A247468.

import Data.Ratio ((%), numerator, denominator)
a058977 = numerator . until ((== 1) . denominator) f . f . fromIntegral
   where f x = a008472 z % a001221 z
               where z = numerator x + denominator x
-- Reinhard Zumkeller, Aug 29 2014

nxt[n_]:=Module[{s=Numerator[n]+Denominator[n]},Total[Transpose[ FactorInteger[ s]][[1]]]/PrimeNu[s]]; Table[NestWhile[nxt,nxt[n],!IntegerQ[#]&],{n,90}] (* Harvey P. Dale, Mar 15 2013 *)

f2(p,q) = my(f=factor(p+q)[,1]~); vecsum(f)/#f;
f1(r) = f2(numerator(r), denominator(r));
loop(list) = {my(v=Vecrev(list)); for (i=2, #v, if (v[i] == v[1], return(1)););}
a(n) = {my(ok=0, m=f2(n,1), list=List()); while(denominator(m) != 1, m = f1(m); listput(list, m); if (loop(list), return (0));); return(m);} \\ Michel Marcus, Feb 09 2022

More terms from Matthew Conroy, Apr 18 2001

A059175 For a rational number p/q let f(p/q) = p*q divided by the sum of digits of p and q; 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

0, 66, 66, 462, 180, 66, 31395, 714, 72, 9, 5, 15, 3, 36, 42, 39, 2, 9, 45, 462, 12, 12, 90, 3703207920, 1692600, 84, 234, 27, 3043425, 74613, 6, 7930296, 264, 4290, 510, 315, 315, 73302369360, 1155, 3, 8, 239872017, 6, 4386, 1989, 18, 17740866, 499954980

Offset: 0

Floor van Lamoen, Jan 15 2001

			3/1 -> 3/4 -> 12/7 -> 84/10=42/5 -> 210/11 -> 2310/5 = 462 so a(3)=462.
84/1 -> 84/13 -> 273/4 -> 273/4 -> ...  so a(84) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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

Cf. A007953, A058971, A214866.

import Data.Ratio ((%), numerator, denominator)
a059175 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))
-- Reinhard Zumkeller, Mar 11 2013

f[Rational[p_, q_]] := p*q/(Total[ IntegerDigits[p]] + Total[ IntegerDigits[q]]); f[n_Integer] := n/(1 + Total[ IntegerDigits[n]]); a[n_] := If[ IntegerQ[ r = NestWhile[f, n, Not[#1 == #2 || #1 != #2 && IntegerQ[#2]]&, 2]], r, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 03 2013 *)

f2(p,q) = p*q/(sumdigits(p)+sumdigits(q));
f1(r) = f2(numerator(r), denominator(r));
loop(list) = {my(v=Vecrev(list)); for (i=2, #v, if (v[i] == v[1], return(1)););}
a(n) = {if (n==0, return(0)); my(ok=0, m=f2(n,1), list=List()); while(denominator(m) != 1, m = f1(m); listput(list, m); if (loop(list), return (0));); return(m);} \\ Michel Marcus, Feb 09 2022

a(A214866(n)) = 0. - Reinhard Zumkeller, Mar 11 2013

Corrected and extended by Naohiro Nomoto, Jul 20 2001

A058988 For a rational number p/q let f(p/q) = p*q divided by number of divisors of p+q; 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, 2, 30, 3, 14, 12, 18, 5, 33, 6, 26, 21, 3, 8, 51, 9, 38, 5, 28, 11, 92, 8, 50, 0, 9, 14, 116, 15, 93, 8, 66, 17, 105, 18, 74, 0, 156, 20, 492, 21, 86, 22, 60, 23, 0, 16, 147, 0, 17, 26, 212, 27, 330, 14, 114, 29, 354, 30, 61, 186, 9, 16, 260, 33, 201, 17, 138, 35, 426

Offset: 1

N. J. A. Sloane, Jan 17 2001

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

Cf. A058972, A058971, A058977.

Cf. A000005.

import Data.Ratio ((%), numerator, denominator)
a058988 n = numerator $ fst $
  until ((== 1) . denominator . fst) f $ f (fromIntegral n, []) where
  f (x, ys) = if y `elem` ys then (0, []) else (y, y:ys) where
   y = numerator x * denominator x % a000005 (numerator x + denominator x)
-- Reinhard Zumkeller, Aug 29 2014

f2(p,q) = p*q/numdiv(p+q);
f1(r) = f2(numerator(r), denominator(r));
loop(list) = {my(v=Vecrev(list)); for (i=2, #v, if (v[i] == v[1], return(1)););}
a(n) = {my(ok=0, m=f2(n,1), list=List()); while(denominator(m) != 1, m = f1(m); listput(list, m); if (loop(list), return (0));); return(m);} \\ Michel Marcus, Feb 09 2022

More terms from Naohiro Nomoto, Jul 20 2001

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 *)

cp's OEIS Frontend

A058972 For a rational number p/q let f(p/q) = sum of aliquot divisors of p+q divided by number of divisors of p+q; sequence gives numbers k such that, starting at k/1 and iterating f, an integer is eventually reached.

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A058977 For a rational number p/q let f(p/q) = sum of distinct prime factors (A008472) of p+q divided by number of distinct prime factors (A001221) of p+q; 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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A059175 For a rational number p/q let f(p/q) = p*q divided by the sum of digits of p and q; 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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A058988 For a rational number p/q let f(p/q) = p*q divided by number of divisors of p+q; 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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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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