A003001 -id:A003001 - cp's OEIS frontend

A121105 Trajectory of 679 under "x -> product of digits of x" map.

679, 378, 168, 48, 32, 6

Offset: 1

N. J. A. Sloane, Aug 12 2006

This is the smallest number of persistence 5 (cf. A003001).

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Most[FixedPointList[Times@@IntegerDigits[#]&,679,7]] (* Harvey P. Dale, Sep 13 2024 *)

Edited by Charles R Greathouse IV, Aug 05 2010

A121110 Trajectory of 3778888999 under "x -> product of digits of x" map.

Original entry on oeis.org

3778888999, 438939648, 4478976, 338688, 27648, 2688, 768, 336, 54, 20, 0

Offset: 1

N. J. A. Sloane, Aug 12 2006

This is the smallest number of persistence 10 (cf. A003001).

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Cf. A003001.

Most[FixedPointList[Times@@IntegerDigits[#]&,3778888999]] (* Harvey P. Dale, Jun 08 2017 *)

Edited by Charles R Greathouse IV, Aug 05 2010

A121111 Trajectory of 277777788888899 under "x -> product of digits of x" map.

Original entry on oeis.org

277777788888899, 4996238671872, 438939648, 4478976, 338688, 27648, 2688, 768, 336, 54, 20, 0

Offset: 1

N. J. A. Sloane, Aug 12 2006

This is the smallest number of persistence 11 (cf. A003001). It is conjectured that no number has persistence greater than 11.

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Cf. A003001.

NestWhileList[Times@@IntegerDigits[#]&,277777788888899,#>0&] (* Harvey P. Dale, Nov 25 2022 *)

a(n)=A121110(n-1), n>=3. [From R. J. Mathar, Aug 07 2008]

Edited by Charles R Greathouse IV, Aug 05 2010

A131809 Multiplicative persistence of Catalan numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 3, 1, 5, 4, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jul 18 2007

After the 57th terms all the numbers have some digits equal to zero thus the persistence is equal to 1.

			Catalan number 429 -> 4*2*9=72 -> 7*2=14 -> 1*4=4 thus persistence is 3

Cf. A003001, A006050, A000108.

P:=proc(n) local i,k,w,ok,cont; for i from 0 by 1 to n do k:=(2*i)!/(i!*(i+1)!); w:=1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(100);

A131810 Additive persistence of Catalan numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 4, 3, 2, 3

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jul 18 2007

			Catalan number 429 -> 4+2+9=15 -> 1+5=6 thus persistence is 2

Cf. A003001, A006050, A000108, A131809.

with(numtheory): with(combinat): P:=proc(n) local a,t; t:=0; a:=(2*n)!/(n!*(n+1)!); while a>9 do t:=t+1; a:=convert(convert(a,base,10),`+`); od; t;
end: seq(P(i),i=0..10^2);

Corrected entries and Maple code by Paolo P. Lava, Dec 19 2017

A176762 Smallest number that takes n steps to reach a cycle under iteration of sum-of-squares-of-digits map.

Original entry on oeis.org

1, 10, 13, 23, 19, 7, 356, 4, 2, 11, 15, 5, 3, 14, 45, 36, 6, 112, 269, 15999

Offset: 0

Robert G. Wilson v, Apr 25 2010

Cf. A001273, A094406, A045646, A003001.

f[n_] := Plus @@ (IntegerDigits[n]^2); t = Table[0, {25}]; k = 1; While[k < 150000001, a = Length@ NestWhileList[f, k, UnsameQ@## &, All] - 1; If[a < 25 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]

a(n) = minimum of either A001273(n) or A094406(n+3).

A201991 Smallest palindrome which has multiplicative persistence n.

Original entry on oeis.org

0, 11, 44, 55, 77, 868, 69996, 2683862, 6783876, 268969862, 37889398873, 477788989887774

Offset: 0

Arkadiusz Wesolowski, Dec 07 2011

Probably finite.

			0 has persistence 0.
11 -> 1 has persistence 1.
44 -> 16 -> 6 has persistence 2.
55 -> 25 -> 10 -> 0 has persistence 3.
77 -> 49 -> 36 -> 18 -> 8 has persistence 4.
868 -> 384 -> 96 -> 54 -> 20 -> 0 has persistence 5.

Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for sequences related to palindromes

Cf. A031346, A003001, A002113.

lst = {}; int[n_] := IntegerDigits[n]; n = 0; Do[While[True, s = Length@int[n]; r = PadRight[int[n], 2*s, Reverse@int[n]]; If[s > 1, r = Drop[r, {s}]]; p = k = FromDigits[r]; c = 0; While[k > 9, k = Times @@ int[k]; c++]; If[c == l, Break[]]; n++]; AppendTo[lst, p], {l, 0, 10}]; lst (* Arkadiusz Wesolowski, Jul 05 2012 *)

A336383 a(n) is the smallest number such that, with f(x) = x - (the product of the digits of x), f(a(n)) reaches a fixed point after n iterations.

Original entry on oeis.org

0, 1, 21, 31, 42, 52, 73, 81, 319, 391, 463, 583, 2911, 3667, 6451, 8793, 9927, 237126, 254158, 278393, 2561363, 9398143, 9431623, 9951265, 83543869, 83896381, 83935261, 2843233127, 2847297383, 2853748583, 2885762663, 266998137657, 685718563667, 688373877587

Offset: 0

Alon Ran, Jul 19 2020

A fixed point occurs once the function returns a number that contains the digit 0. After that, the product of the digits will be 0, and so subtracting it from the number will be idempotent.

This sequence is conceptually similar to A003001, though unlike the latter, it is probably infinite.

			a(9) = 391 because:
   1: 391 - 3*9*1 = 364
   2: 364 - 3*6*4 = 292
   3: 292 - 2*9*2 = 256
   4: 256 - 2*5*6 = 196
   5: 196 - 1*9*6 = 142
   6: 142 - 1*4*2 = 134
   7: 134 - 1*3*4 = 122
   8: 122 - 1*2*2 = 118
   9: 118 - 1*1*8 = 110
After iteration 9, the function becomes idempotent:
  10: 110 - 1*1*0 = 110
  11: 110 - 1*1*0 = 110
  12: 110 - 1*1*0 = 110
  ...
Additionally, 391 is the smallest number with this property. Thus, it is a(9).

Cf. A003001, A070565.

nmax = 20; tab = ConstantArray[Null, nmax];
For[k = 0, k <= 1000000, k++,
 l=Length@  NestWhileList[#-Times @@ IntegerDigits[#] &,k,UnsameQ[##] &, 2]-2;
If[tab[[l+1]] == Null, tab[[l+1]] = k]]; tab  (* Robert Price, Sep 13 2020 *)

f(m) = m - vecprod(digits(m)) + (m==0);
lista(nn) = {my(c, m, t); for(k=0, nn, c=0; m=k; while(m!=(m=f(m)), c++); if(c==t, print1(k, ", "); t++)); } \\ Jinyuan Wang, Aug 14 2020

def f(x):
    prod = 1
    for digit in str(x):
        prod *= int(digit)
    return x - prod
def a(n):
    i = 0
    iteration = 0
    while iteration != n:
        i += 1
        j = i
        iteration = 0
        new_j = f(j)
        while j != new_j:
            iteration += 1
            j = new_j
            new_j = f(j)
    return i

a(27)-a(30) from Jinyuan Wang, Aug 14 2020

a(31)-a(33) added by Michael S. Branicky, Aug 29 2020

A336682 a(n) is the number of iterations needed to reach a fixed point starting with n and repeatedly applying f(x) = x - (the product of the digits of x).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 3, 3, 3, 3, 1, 2, 2, 2, 2, 0, 3, 4, 4, 3, 3, 3, 2, 2, 2, 0, 4, 5, 3, 4, 1, 3, 3, 2, 2, 0, 2, 1, 4, 1, 2, 1, 3, 1, 2, 0, 2, 3, 6, 4, 1, 4, 3, 3, 2, 0, 7, 2, 3, 6, 4

Offset: 0

Robert Price, Sep 13 2020

			a(42) = 4 because:
1: 42 - 4*2 = 34
2: 34 - 3*4 = 22
3: 22 - 2*2 = 18
4: 18 - 1*8 = 10
5: 10 - 1*0 = 10

Robert Price, Table of n, a(n) for n = 0..1000

Cf. A003001, A070565, A336383.

Table[Length@  NestWhileList[# - Times @@ IntegerDigits[#] &, n, UnsameQ[##] &,
2] - 2, {n, 0, 85}];(* Robert Price, Sep 13 2020 *)

A067670 Smallest prime of additive persistence n.

Original entry on oeis.org

2, 11, 19, 199, 29998999999999999999999

Offset: 0

Shyam Sunder Gupta, Feb 23 2002

a(5) >= 29*10^2222222222222222222221-1, the next number of additive persistence 5 after A006050(5). (a(5) is not equal to A006050(5) because that number is divisible by 313.) - Pontus von Brömssen, Oct 17 2023

			a(2) = 19, 19 -> 10 -> 1, so 2 summation steps are required to reach a single-digit number.

H. J. Hindin, The additive persistence of a number, J. Rec. Math., 7 (No. 2, 1974), 134-135.

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Additive Persistence

Cf. A003001, A045646, A006050.

a(n) >= A006050(n). - Pontus von Brömssen, Oct 17 2023

cp's OEIS Frontend

A121105 Trajectory of 679 under "x -> product of digits of x" map.

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A121110 Trajectory of 3778888999 under "x -> product of digits of x" map.

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A121111 Trajectory of 277777788888899 under "x -> product of digits of x" map.

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A131809 Multiplicative persistence of Catalan numbers.

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A131810 Additive persistence of Catalan numbers.

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A176762 Smallest number that takes n steps to reach a cycle under iteration of sum-of-squares-of-digits map.

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A201991 Smallest palindrome which has multiplicative persistence n.

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A336383 a(n) is the smallest number such that, with f(x) = x - (the product of the digits of x), f(a(n)) reaches a fixed point after n iterations.

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