A046514 -id:A046514 - 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

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

Original entry on oeis.org

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)])

A046505 Primes with multiplicative persistence value 5.

Original entry on oeis.org

769, 967, 1697, 2777, 3637, 3673, 4483, 6197, 6337, 6373, 6719, 6733, 6779, 6791, 6917, 6971, 6977, 7559, 7691, 7727, 8443, 8681, 8861, 9677, 9767, 12379, 12739, 12973, 13297, 13367, 13729, 13763, 14779, 14797, 14843, 17239, 17293, 17497

Offset: 1

Patrick De Geest, Sep 15 1998

			769 -> [ 378 ][ 168 ][ 48 ][ 32 ][ 6 ] -> one digit in five steps.

Daniel Mondot, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A000040 and A046514.

Cf. A046500.

mp5Q[n_]:=Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>9&]]==6; Select[ Prime[Range[2100]],mp5Q] (* Harvey P. Dale, Dec 30 2019 *)

A199995 Composite numbers whose multiplicative persistence is 5.

Original entry on oeis.org

679, 688, 697, 796, 868, 886, 976, 1679, 1688, 1769, 1796, 1868, 1886, 1967, 1976, 2379, 2388, 2397, 2468, 2486, 2648, 2684, 2688, 2739, 2793, 2838, 2846, 2864, 2868, 2883, 2886, 2937, 2973, 3279, 3288, 3297, 3367, 3376, 3448, 3484, 3488, 3729, 3736, 3763

Offset: 1

Jaroslav Krizek, Nov 13 2011

Complement of A046505 with respect to A046514.

			679 -> [ 378 ][ 168 ][ 48 ][ 32 ][ 6 ] -> one digit in five steps.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046505 (primes whose multiplicative persistence is 5).

persistence[n_] := Module[{cnt = 0, k = n}, While[k > 9, cnt++; k = Times @@ IntegerDigits[k]]; cnt]; Select[Range[4000], !PrimeQ[#] && persistence[#] == 5 &] (* T. D. Noe, Nov 23 2011 *)
mp5Q[n_]:=CompositeQ[n]&&Length[NestWhileList[ Times@@IntegerDigits[ #]&,n,#>9&]] == 6; Select[Range[4000],mp5Q] (* Harvey P. Dale, Apr 23 2022 *)

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A046510 Numbers with multiplicative persistence value 1.

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

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A046505 Primes with multiplicative persistence value 5.

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A199995 Composite numbers whose multiplicative persistence is 5.

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