A031346 -id:A031346 - cp's OEIS frontend

A350184 Numbers of multiplicative persistence 5 which are themselves the product of digits of a number.

2688, 18816, 26244, 98784, 222264, 262144, 331776, 333396, 666792, 688128, 1769472, 2939328, 3687936, 4214784, 4917248, 13226976, 19361664, 38118276, 71663616, 111476736, 133413966, 161414428, 169869312, 184473632, 267846264, 368947264, 476171136, 1783627776

Offset: 1

Daniel Mondot, Dec 18 2021

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 5.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181....
Equivalently:
This sequence consists of all numbers A007954(k) such that A031346(k) = 6.
These are the numbers k in A002473 such that A031346(k) = 5.
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 5 steps.
Postulated to be finite and complete.

			2688 is in this sequence because:
- 2688 goes to a single digit in 5 steps: p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
- p(27648) = p(47628) = 2688, etc.
331776 is in this sequence because:
- 331776 goes to a single digit in 5 steps: p(331776)=2646, p(2646)=288, p(288)=128, p(128)=16, p(16)=6.
- p(914838624) = p(888899) = 331776, etc.

Daniel Mondot, Table of n, a(n) for n = 1..41
Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A002473 and A046514 (all numbers with mp of 5).
Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root).
Cf. A350180, A350181, A350182, A350183, A350185, A350186, A350187 (numbers with mp 1 to 4 and 6 to 10 that are themselves 7-smooth numbers).

mx=10^10;lst=Sort@Flatten@Table[2^i*3^j*5^k*7^l,{i,0,Log[2,mx]},{j,0,Log[3,mx/2^i]},{k,0,Log[5,mx/(2^i*3^j)]},{l,0,Log[7,mx/(2^i*3^j*5^k)]}];
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==5&] (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *)

from math import prod
def hd(n):
    while (n&1) == 0:  n >>= 1
    while (n%3) == 0:  n /= 3
    while (n%5) == 0:  n /= 5
    while (n%7) == 0:  n /= 7
    return(n)
def pd(n): return prod(map(int, str(n)))
def ok(n):
    if hd(n) > 9: return False
    return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and (s := pd(r)) > 9 and pd(s) < 10
print([k for k in range(10,476200000) if ok(k)])

A350185 Numbers of multiplicative persistence 6 which are themselves the product of digits of a number.

Original entry on oeis.org

27648, 47628, 64827, 84672, 134217728, 914838624, 1792336896, 3699376128, 48814981614, 134481277728, 147483721728, 1438916737499136

Offset: 1

Daniel Mondot, Jan 15 2022

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 7.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181....
Equivalently:
This sequence consists of the numbers A007954(k) such that A031346(k) = 7,
These are the numbers k in A002473 such that A031346(k) = 6,
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 6 steps.
Postulated to be finite and complete.
a(13), if it exists, is > 10^20000, and likely > 10^80000.

			27648 is in sequence because:
- 27648 goes to a single digit in 6 steps: p(27648)=2688, p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
- p(338688) = p(168889) = 27648, etc.

Daniel Mondot, Multiplicative Persistence Tree

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046515 (all numbers with mp of 6).

Cf. A350180, A350181, A350182, A350183, A350184, A350186, A350187 (numbers with mp 1 to 5 and 7 to 10 that are themselves 7-smooth numbers).

mx=10^16;lst=Sort@Flatten@Table[2^i*3^j*5^k*7^l,{i,0,Log[2,mx]},{j,0,Log[3,mx/2^i]},{k,0,Log[5,mx/(2^i*3^j)]},{l,0,Log[7,mx/(2^i*3^j*5^k)]}];
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==6&] (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *)

#this program may take 91 minutes to produce the first 8 members.
from math import prod
def hd(n):
    while (n&1) == 0:  n >>= 1
    while (n%3) == 0:  n /= 3
    while (n%5) == 0:  n /= 5
    while (n%7) == 0:  n /= 7
    return(n)
def pd(n): return prod(map(int, str(n)))
def ok(n):
    if hd(n) > 9: return False
    return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and (s := pd(r)) > 9 and (t := pd(s)) > 9 and pd(t) < 10
print([k for k in range(10,3700000000) if ok(k)])

A350187 Numbers of multiplicative persistence 8 which are themselves the product of digits of a number.

Original entry on oeis.org

4478976, 784147392, 19421724672, 249143169618, 717233481216

Offset: 1

Daniel Mondot, Jan 30 2022

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 9.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181.
Equivalently:
This sequence consists of all numbers A007954(k) such that A031346(k) = 9.
They are the numbers k in A002473 such that A031346(k) = 8.
Or they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 8 steps.
Postulated to be finite and complete.
a(6), if it exists, is > 10^20000, and likely > 10^171000.

			4478976 is in this sequence because:
- 4478976 goes to a single digit in 8 steps: 4478976 -> 338688 -> 27648 -> 2688 -> 768 -> 336 -> 54 -> 20 -> 0;
- p(438939648) = p(231928233984) = 4478976.

Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A002473 and A046517.
Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046517 (all numbers with mp of 8).
Cf. A350180, A350181, A350182, A350183, A350184, A350185, A350186 (numbers with mp 1 to 7 and 9 to 10 that are themselves 7-smooth numbers).

A064871 The minimal number which has multiplicative persistence 7 in base n.

Original entry on oeis.org

1409794, 68889, 38200, 17902874277, 1494, 2532, 19526, 15838, 1101, 15820, 943, 2674, 2118, 3275, 412, 3310, 1593, 440, 478, 2036, 456, 713, 738, 633, 658, 705, 907, 643, 803, 641, 653, 797, 484, 991, 814, 877, 1079, 767, 840, 575, 930, 843, 710, 880

Offset: 9

Sascha Kurz, Oct 08 2001

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit. a(7) = 686285, a(8) seems not to exist.

			a(9) = 1409794 because the persistence of 1409794 is 7.

M. R. Diamond and D. D. Reidpath, A counterexample to a conjecture of Sloane and Erdos, J. Recreational Math., 1998 29(2), 89-92.
Sascha Kurz, Persistence in different bases
T. Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
C. Rivera, Minimal prime with persistence p
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Index entries for linear recurrences with constant coefficients, order 5041.

Cf. A003001, A031346, A064867, A064868, A064869, A064870, A064872.

a(n) = 8*n-[n/5040] for n > 5039.

Corrected by R. J. Mathar, Nov 02 2007

A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.

Original entry on oeis.org

338688, 826686, 2239488, 3188646, 6613488, 14224896, 3416267673274176, 6499837226778624

Offset: 1

Daniel Mondot, Jan 15 2022

The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 8.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181....
Equivalently:
This sequence consists of the numbers A007954(k) such that A031346(k) = 8,
These are the numbers k in A002473 such that A031346(k) = 7,
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 7 steps.
Postulated to be finite and complete.
a(9), if it exists, is > 10^20000, and likely > 10^119000.

			338688 is in this sequence because:
- 338688 goes to a single digit in 7 steps: p(338688) = 27648, p(27648) = 2688, p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
- p(4478976) = p(13477889) = 338688, etc.

Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046516 (all numbers with mp of 7).

Cf. A350180, A350181, A350182, A350183, A350184, A350185, A350187 (numbers with mp 1 to 6 and 8 to 10 that are themselves 7-smooth numbers).

mx=10^16;lst=Sort@Flatten@Table[2^i*3^j*5^k*7^l,{i,0,Log[2,mx]},{j,0,Log[3,mx/2^i]},{k,0,Log[5,mx/(2^i*3^j)]},{l,0,Log[7,mx/(2^i*3^j*5^k)]}];
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==7&]  (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *)

A087472 Number of iterations required for the function f(n) to reach a single digit, where f(n) is the product of the two numbers formed from the alternating digits of n.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy and Paul D. Hanna, Sep 11 2003

A087471(n) gives the final digit reached by successive iterations of Murthy's function, f(n). A087473(n) gives the smallest number that requires n iterations of Murthy's function to reach a single digit. The n-th row of triangle A087474 gives the n successive iterations of Murthy's function on A087473(n).

Differs from A031346 first at n=110. [From R. J. Mathar, Sep 11 2008]

			a(1234)= 3 since f(1234)=13*24=312, f(312)=32*1=32 and
f(32)=3*2=6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A087471, A087473, A087474.

murthy:= proc(n) local L,d;
  L:= convert(n,base,10);
  d:= nops(L);
  add(L[2*i+1]*10^i,i=0..(d-1)/2)*add(L[2*i+2]*10^i,i=0..(d-2)/2)
end proc:
A087472:= proc(n) option remember;
  if n < 10 then  0 else 1+procname(murthy(n)) fi
end proc:
map(A087472, [$1..200]); # Robert Israel, Feb 14 2017

A239427 Numbers such that additive and multiplicative persistences coincide.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 37, 38, 40, 41, 42, 46, 48, 50, 51, 56, 58, 60, 61, 64, 65, 67, 70, 71, 73, 76, 80, 81, 82, 83, 84, 85, 90, 92, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Arkadiusz Wesolowski, Mar 19 2014

Numbers n for which A031286(n) = A031346(n).

			28 -> 10 -> 1 has additive persistence 2. 28 -> 16 -> 6 has multiplicative persistence 2. 28 is therefore in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..66 from Arkadiusz Wesolowski)
Eric Weisstein's World of Mathematics, Additive Persistence
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Supersequence of A239480. Cf. A031286, A031346, A064702.

for(n=0, 106, v=n; a=0; while(n>9, a++; n=sumdigits(n)); n=v; m=0; while(n>9, m++; d=digits(n); n=prod(k=1, #d, d[k])); n=v; if(a==m, print1(n, ", ")));

from math import prod
def A031286(n):
    ap = 0
    while n > 9: n, ap = sum(map(int, str(n))), ap+1
    return ap
def A031346(n):
    mp = 0
    while n > 9: n, mp = prod(map(int, str(n))), mp+1
    return mp
def ok(n): return A031286(n) == A031346(n)
print([k for k in range(107) if ok(k)]) # Michael S. Branicky, Sep 17 2022

A014553 Maximal multiplicative persistence (or length) of any n-digit number.

Original entry on oeis.org

1, 4, 5, 6, 7, 7, 8, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Eric W. Weisstein

The "persistence" or "length" of an N-digit decimal number is the number of times one must iteratively form the product of its digits until one obtains a one-digit product (For another definition see A003001.)

For all other n<2530, a(n)=11 because sequence is nondecreasing and a number with multiplicative persistence 12 must have more than 2530 digits. - Sascha Kurz, Mar 24 2002

			168889 is not in A003001 because a(6) = a(5) = 7.

Gottlieb, A. J. Problems 28-29 in "Bridge, Group Theory and a Jigsaw Puzzle." Techn. Rev. 72, unpaginated, Dec. 1969.
Gottlieb, A. J. Problem 29 in "Integral Solutions, Ladders and Pentagons." Techn. Rev. 72, unpaginated, Apr. 1970.

Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 56
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Cf. A003001, A031346, A035927.

Corrected by N. J. A. Sloane, Nov 1995

More terms from John W. Layman, Mar 19 2002

A031348 2-multiplicative persistence: number of iterations of "multiply 2nd powers of digits" needed to reach 0 or 1.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

From Mohammed Yaseen, Nov 08 2022: (Start)
Is 7 the maximal 2-multiplicative persistence?
Are A199986 the only numbers whose 2-multiplicative persistence is 7?
These hold true for n up to 10^9. (End)

			a(14) = 6 because
14 -> 1^2 * 4^2 = 16;
16 -> 1^2 * 6^2 = 36;
36 -> 3^2 * 6^2 = 324;
324 -> 3^2 * 2^2 * 4^2 = 576;
576 -> 5^2 * 7^2 * 6^2 = 44100;
44100 -> 0 => the trajectory is 14 -> 16 -> 36 -> 324 -> 576 -> 44100 -> 0 with 6 iterations. - _Michel Lagneau_, May 22 2013

M. Gardner, Fractal Music, Hypercards and More Mathematical Recreations from Scientific American, Persistence of Numbers, pp. 120-1; 186-7, W. H. Freeman, NY, 1992.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
M. R. Diamond, Multiplicative persistence base 10: some new null results.
N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A031346.

m2pd[n_]:=Length[NestWhileList[Times@@(IntegerDigits[#]^2)&,n,#>1&]]-1; Array[m2pd,100] (* Harvey P. Dale, Apr 19 2020 *)

f(n) = my(d=digits(n)); prod(k=1, #d, d[k]^2);
a(n) = if (n==1, 0, my(nb=1); while(((new = f(n)) > 1), n = new; nb++); nb); \\ Michel Marcus, Jun 13 2018

from math import prod
from itertools import count, islice
def f(n): return prod(map(lambda x: x*x, map(int, str(n))))
def a(n):
    c = 0
    while n not in {0, 1}: n, c = f(n), c+1
    return c
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Oct 13 2022

A064700 Numbers k that are divisible by the multiplicative digital root of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 26, 34, 35, 48, 62, 64, 72, 75, 84, 88, 111, 112, 115, 126, 132, 134, 135, 136, 144, 162, 168, 172, 174, 175, 176, 186, 192, 212, 216, 228, 232, 246, 248, 264, 276, 278, 282, 288, 312, 314, 315, 322, 355, 364, 376, 378

Offset: 1

Santi Spadaro, Oct 12 2001

No term has 0 as one of its digits.

The only primes in the sequence are {2, 3, 5, 7, 11} and any other prime that has only 1s as digits, such as 1111111111111111111.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A031346, A007954, A031347, A052382.

a064700 n = a064700_list !! (n-1)
a064700_list = filter f [1..] where
   f x = mdr > 0 && x `mod` mdr == 0 where mdr = a031347 x
-- Reinhad Zumkeller, Sep 22 2011

mdr[n_] := FixedPoint[ Times @@ IntegerDigits[#] &, n]; Select[ Range[400], (m = mdr[#]; m > 0 && Mod[#, m] == 0) &] (* Jean-François Alcover, Nov 30 2011 *)
dvsbQ[n_]:=Mod[n,NestWhile[Times@@IntegerDigits[#]&,n,#>9&]/.(0->Pi)]==0; Select[Range[ 500], dvsbQ] (* Harvey P. Dale, Aug 09 2023 *)

cp's OEIS Frontend

A350184 Numbers of multiplicative persistence 5 which are themselves the product of digits of a number.

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A350185 Numbers of multiplicative persistence 6 which are themselves the product of digits of a number.

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A350187 Numbers of multiplicative persistence 8 which are themselves the product of digits of a number.

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A064871 The minimal number which has multiplicative persistence 7 in base n.

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A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.

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A087472 Number of iterations required for the function f(n) to reach a single digit, where f(n) is the product of the two numbers formed from the alternating digits of n.

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A239427 Numbers such that additive and multiplicative persistences coincide.

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A014553 Maximal multiplicative persistence (or length) of any n-digit number.

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