A046512 -id:A046512 - cp's OEIS frontend

A046510 Numbers with multiplicative persistence value 1.

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 30, 31, 32, 33, 40, 41, 42, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 130, 131, 132, 133

Offset: 1

Patrick De Geest, Sep 15 1998

Numbers 0 to 9 have a multiplication persistence of 0, not 1. - Daniel Mondot, Mar 12 2022

			24 -> 2 * 4 = [ 8 ] -> one digit in one step.

Daniel Mondot, Table of n, a(n) for n = 1..10000
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, A014120, A046501.

Numbers with multiplicative persistence m: this sequence (m=1), A046511 (m=2), A046512 (m=3), A046513 (m=4), A046514 (m=5), A046515 (m=6), A046516 (m=7), A046517 (m=8), A046518 (m=9), A352531 (m=10), A352532 (m=11).

Select[Range[10, 121], IntegerLength[Times @@ IntegerDigits[#]] <= 1 &] (* Jayanta Basu, Jun 26 2013 *)

isok(n) = my(d=digits(n)); (#d > 1) && (#digits(prod(k=1, #d, d[k])) <= 1); \\ Michel Marcus, Apr 12 2018 and Mar 13 2022

from math import prod
def ok(n): return n > 9 and prod(map(int, str(n))) < 10
print([k for k in range(134) if ok(k)]) # Michael S. Branicky, Mar 13 2022

Incorrect terms 0 to 9 removed by Daniel Mondot, Mar 12 2022

A350182 Numbers of multiplicative persistence 3 which are themselves the product of digits of a number.

Original entry on oeis.org

49, 75, 96, 98, 147, 168, 175, 189, 196, 288, 294, 336, 343, 392, 448, 486, 648, 672, 729, 784, 864, 882, 896, 972, 1344, 1715, 1792, 1944, 2268, 2744, 3136, 3375, 3888, 3969, 7938, 8192, 9375, 11664, 12288, 12348, 13824, 14336, 16384, 16464, 17496, 18144

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 4.
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) = 4,
These are the numbers k in A002473 such that A031346(k) = 3,
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 3 steps.
Postulated to be finite and complete.
Let p(n) be the product of all the digits of n.
The multiplicative persistence of a number mp(n) is the number of times you need to apply p() to get to a single digit.
For example:
mp(1) is 0 since 1 is already a single-digit number.
mp(10) is 1 since p(10) = 0, and 0 is a single digit, 1 step.
mp(25) is 2 since p(25) = 10, p(10) = 0, 2 steps.
mp(96) is 3 since p(96) = 54, p(54) = 20, p(20) = 0, 3 steps.
mp(378) is 4 since p(378) = 168, p(168) = 48, p(48) = 32, p(32) = 6, 4 steps.
There are infinitely many numbers n such that mp(n)=4. But for each n with mp(n)=4, p(n) is a number included in this sequence, and this sequence is likely finite.
This sequence lists p(n) such that mp(n) = 4, or mp(p(n)) = 3.

			49 is in this sequence because:
- 49 goes to a single digit in 3 steps: p(49) = 36, p(36) = 18, p(18) = 8.
- p(77) = p(177) = p(717) = p(771) = 49, etc.
75 is in this sequence because:
- 75 goes to a single digit in 3 steps: p(75) = 35, p(35) = 15, p(15) = 5.
- p(355) = p(535) = p(1553) = 75, etc.

Daniel Mondot, Table of n, a(n) for n = 1..387
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Multiplicative Persistence Tree

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046512 (all numbers with mp of 3).

Cf. A350180, A350181, A350183, A350184, A350185, A350186, A350187 (numbers with mp 0, 1 and 3 to 10 that are themselves 7-smooth numbers).

A046503 Primes with multiplicative persistence value 3.

Original entry on oeis.org

47, 59, 79, 89, 97, 139, 149, 157, 179, 193, 197, 227, 229, 239, 263, 283, 293, 337, 347, 353, 359, 367, 373, 383, 389, 419, 433, 443, 449, 463, 479, 487, 491, 499, 571, 577, 593, 619, 643, 661, 673, 683, 691, 719, 733, 743, 751, 757, 797, 823, 829, 839

Offset: 1

Patrick De Geest, Sep 15 1998

			47 -> 4 * 7 -> [ 28 ] -> 2 * 8 -> [ 16 ] -> 1 * 6 -> [ 6 ] -> one digit in three steps.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A046500, A046512.

filter:= proc(n) local L;
  if not isprime(n) then return false fi;
  L:= convert(convert(n,base,10),`*`); if L < 10 then return false fi;
  L:= convert(convert(L,base,10),`*`); if L < 10 then return false fi;
  L:= convert(convert(L,base,10),`*`); evalb(L < 10)
end proc:
select(filter, [seq(i,i=11..1000,2)]); # Robert Israel, Jun 05 2018

pr3Q[n_] := Length[NestWhileList[Times @@ IntegerDigits[#] &, n, # > 9 &]] == 4; Select[Prime[Range[147]], pr3Q] (* Jayanta Basu, Jun 26 2013 *)

A199993 Composite numbers whose multiplicative persistence is 3.

Original entry on oeis.org

39, 49, 55, 57, 66, 68, 69, 74, 75, 78, 86, 87, 88, 93, 94, 95, 96, 98, 147, 155, 159, 166, 168, 169, 174, 175, 178, 186, 187, 188, 189, 194, 195, 196, 198, 236, 238, 246, 247, 248, 249, 264, 266, 267, 272, 274, 276, 279, 284, 288, 289, 292, 294, 297, 298, 299

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A046503 with respect to A046512.

			147 -> 1 * 4 * 7 -> [ 28 ] -> 2 * 8 -> [ 16 ] -> 1 * 6 -> [ 6 ] -> one digit in three steps.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Cf. A046503 (primes whose multiplicative persistence is 3).

Select[Range[300],CompositeQ[#]&&Length[NestWhileList[Times@@ IntegerDigits[ #]&,#,#>9&]] == 4&] (* Harvey P. Dale, Mar 29 2023 *)

{ A046512 } minus { A046503 }.

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A046510 Numbers with multiplicative persistence value 1.

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A350182 Numbers of multiplicative persistence 3 which are themselves the product of digits of a number.

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A046503 Primes with multiplicative persistence value 3.

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A199993 Composite numbers whose multiplicative persistence is 3.

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