A003001 -id:A003001 - cp's OEIS frontend

A028843 Numbers whose iterated product of digits is a prime.

2, 3, 5, 7, 12, 13, 15, 17, 21, 26, 31, 34, 35, 37, 43, 51, 53, 57, 62, 71, 73, 75, 112, 113, 115, 117, 121, 126, 131, 134, 135, 137, 143, 151, 153, 157, 162, 171, 173, 175, 211, 216, 223, 232, 261, 278, 279, 287, 297, 299, 311, 314, 315, 317, 322, 341, 351, 355

Offset: 1

N. J. A. Sloane

			For 53, the product of digits is 5 * 3 = 15, iterated to 1 * 5 = 5, which is a prime, so 53 is in the sequence.
For 54, the product of digits is 5 * 4 = 20, iterated to 2 * 0 = 0, which is not prime, so 54 is not in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003001, A028835, A028842, A031347, A333955.

iterDigitProd[n_] := NestWhile[Times@@IntegerDigits[#] &, n, # > 9 &]; Select[Range[355], PrimeQ[iterDigitProd[#]] &] (* Jayanta Basu, Jun 02 2013 *)

def iterDigitProd(n: Int): Int = n.toString.length match {
  case 1 => n
  case  => iterDigitProd(n.toString.toCharArray.map( - 48).scanRight(1)( * ).head)
}
(1 to 400).filter(n => List(2, 3, 5, 7).contains(iterDigitProd(n))) // Alonso del Arte, Apr 11 2020

More terms from Patrick De Geest, Jun 15 1999

Corrected by Franklin T. Adams-Watters, Jan 17 2007

A133048 Powerback(n): reverse the decimal expansion of n, drop any leading zeros, then apply the powertrain map of A133500 to the resulting number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 1, 4, 9, 16, 25, 36, 49, 64, 81, 3, 1, 8, 27, 64, 125, 216, 343, 512, 729, 4, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 5, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 6, 1, 64, 729, 4096, 15625, 46656, 117649

Offset: 0

J. H. Conway and N. J. A. Sloane, Dec 31 2007

a(A221221(n)) = A133500(A221221(n)) = A222493(n). - Reinhard Zumkeller, May 27 2013

			E.g. 240 -> (0)42 -> 4^2 = 16; 12345 -> 54321 -> 5^4*3^2*1 = 5625.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A131571 (fixed points), A133059 and A133134 (records); A133500 (powertrain).
Cf. A133144 (length of trajectory), A031346 and A003001 (persistence).
Cf. A031298.

a133048 0 = 0
a133048 n = train $ dropWhile (== 0) $ a031298_row n where
   train []       = 1
   train [x]      = x
   train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013

powerback:=proc(n) local a,i,j,t1,t2,t3;
if n = 0 then RETURN(0); fi;
t1:=convert(n, base, 10); t2:=nops(t1);
for i from 1 to t2 do if t1[i] > 0 then break; fi; od:
a:=1; t3:=t2-i+1;
for j from 0 to floor(t3/2)-1 do a := a*t1[i+2*j]^t1[i+2*j+1]; od:
if t3 mod 2 = 1 then a:=a*t1[t2]; fi;
RETURN(a); end;

ptm[n_]:=Module[{idn=IntegerDigits[IntegerReverse[n]]},If[ EvenQ[ Length[idn]],Times@@ (#[[1]]^#[[2]]&/@Partition[idn,2]),(Times@@(#[[1]]^#[[2]]&/@Partition[ Most[ idn],2]))Last[idn]]];Array[ptm,70,0] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 05 2020 *)

A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 40, 42, 50, 60, 70, 80, 81, 90, 100, 105, 108, 112, 120, 140, 150, 160, 180, 200, 210, 240, 250, 270, 280, 300, 320, 350, 360, 400, 405, 420, 450, 480, 490, 500, 504, 540, 560, 600, 630, 640, 700, 720, 750, 800

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 2.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for subsequent sequences A350181..., but not for this sequence (where mp(p(n)) = 1). That is because there are infinitely many numbers that include both an even digit (2, 4, 6 or 8), a 5 and no 0. For these numbers n, p(n) will include a zero and p(p(n)) will be 0.
Equivalently: This sequence contains all numbers A007954(k) such that A031346(k) = 2, and they are the numbers k in A002473 such that A031346(k) = 1.
Or, they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 1 step.

			10 is in this sequence because:
- 10 goes to a single digit in 1 step: p(10) = 0.
- 25, 52, 125, 152, 215, 512, 251, 521, 1125, 1152, 1215, 1512, 1251, 1521, 2115, 5112, 2511, 5211, etc. all lead to 10, i.e., p(25)=10, p(52)=10, etc.
Some of these (25, 125, 512, 1125, 1152, 1215, 1512) are in the next layer of classes, A350181, and the rest are not.
12 is in this sequence because:
- 12 goes to a single digit in 1 step: p(12) = 2.
- 12, 21, 112, 211, 121, 11112, 11211, etc. all lead to 12.
(12, 21 and 112 are in the next layer of classes, A350181, but the rest are not)
14 is in this sequence because:
- 14 goes to a single digit in 1 step: p(14) = 4.
- 27, 72, 127, 172, 217, 712, 271, 721, 12111711, etc. all lead to 14.
(27 and 72 are in the next layer of classes, A350181, the rest are not).

Daniel Mondot, Table of n, a(n) for n = 1..20000
Daniel Mondot, Multiplicative Persistence Tree
Eric Weisstein's World of Mathematics, Multiplicative Persistence

Intersection of A002473 and A046510
Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046510 (all numbers with mp of 1).
Cf. A350181, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 2 to 10 that are themselves 7-smooth numbers).

mp(n)={my(k=0); while(n>=10, k++; n=vecprod(digits(n))); k}
isparent(n)={my(m=0); while(m<>n, m=n; n/=gcd(n,2*3*5*7)); n==1}
isok(n)={mp(n)==1 && isparent(n)} \\ Andrew Howroyd, Dec 20 2021

A064870 The minimal number which has multiplicative persistence 6 in base n.

Original entry on oeis.org

11262, 57596799, 30536, 6788, 4684, 1571, 439, 667, 1964, 683, 218, 857, 264, 278, 353, 393, 227, 382, 344, 311, 319, 307, 283, 417, 422, 381, 485, 436, 349, 431, 436, 449, 421, 469, 327, 575, 598, 483, 539, 413, 511, 517, 534, 641, 611, 609, 476, 479

Offset: 7

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(5)=1811981201171874, a(6) seems not to exist.

			a(13) = 439 because 439 = [2'7'10]->[10'10]->[7'9]->[4'11]->[3'5]->[1'2]->[2] needs 6 steps and no fewer n.

Michael De Vlieger, Table of n, a(n) for n = 7..10000
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 721.

Cf. A003001, A031346, A064867, A064868, A064869, A064871, A064872.

a(n) = 7*n-[n/720] for n > 719.

A064872 The minimal number which has multiplicative persistence 8 in base n.

Original entry on oeis.org

7577, 130883, 596667, 3644381, 2820, 61773, 2752, 5136, 7452, 38631, 2780, 8015, 2996, 542, 8611, 4591, 575, 10586, 2532, 2681, 2764, 1016, 4547, 10151, 1065, 983, 813, 5431, 900, 1255, 983, 5179, 5117, 1190, 982, 1129, 1501, 1491, 1471, 1084

Offset: 13

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)=1086400325525346, a(10)=2677889, a(11)=757074, a(8) and a(9) seem not to exist.

			a(13) = 7577 because 7577 is the fewest number with persistence 8 in base 13.

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 40321.

Cf. A003001, A031346, A064867, A064868, A064869, A064870, A064871.

a(n) = 9*n-[n/40320] for n > 40319.

A070562 Fecundity of n.

Original entry on oeis.org

0, 10, 9, 9, 8, 1, 8, 7, 7, 6, 0, 8, 7, 7, 6, 1, 6, 6, 5, 3, 0, 5, 5, 4, 5, 2, 4, 5, 2, 3, 0, 3, 4, 2, 2, 1, 3, 3, 3, 2, 0, 4, 1, 2, 1, 3, 1, 2, 1, 4, 0, 5, 3, 8, 2, 1, 4, 2, 2, 1, 0, 2, 2, 5, 5, 2, 1, 1, 7, 5, 0, 4, 4, 2, 1, 1, 6, 5, 3, 2, 0, 4, 2, 1, 7, 3, 3, 3, 4, 3, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, May 07 2002

Start with x=n, repeatedly replace x by x + product of digits of x until the product is 0; fecundity = number of steps. a(0) = 0 by convention.

			1 -> 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 -> 102 has fecundity 10.

P. Tougne, Jeux Mathematiques column, Pour La Science (French edition of "Scientific American"), Vol. 82, Aug. 1984, Prob. 6, pp. 101, 104.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A070560. A070561, A031346, A003001, A070061, A070257.

f[ n_ ] := Block[ {a=n,b,c=0}, While[ b=Times@@IntegerDigits[ a ]; b>0, a=a+b; c++ ]; c ]; f[ 0 ]=0; Table[ f[ n ], {n,0,100} ]
f[n_] := Length@ FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 2; Array[f, 105, 0] (* Robert G. Wilson v, Jun 27 2010 *)

prodig(n) = local(s, d); if(n==0, s=0, s=1; while(n>0, d=divrem(n, 10); n=d[1 ]; s=s*d[2 ])); s for(n=0, 92, x=n; c=0; while((d=prodig(x))!=0, c++; x=x+d); print1(c, ", "))

Edited and extended by Klaus Brockhaus, May 08 2002
Clarified the definition of fecundity and improved the Mathematica program. - T. D. Noe, Oct 06 2008
More terms from Robert G. Wilson v, Jun 27 2010

A350181 Numbers of multiplicative persistence 2 which are themselves the product of digits of a number.

Original entry on oeis.org

25, 27, 28, 35, 36, 45, 48, 54, 56, 63, 64, 72, 84, 125, 126, 128, 135, 144, 162, 192, 216, 224, 225, 243, 245, 252, 256, 315, 324, 375, 432, 441, 512, 525, 567, 576, 588, 625, 675, 735, 756, 875, 945, 1125, 1134, 1152, 1176, 1215, 1225, 1296, 1323, 1372

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 3.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for this and subsequent sequences A350182....
Equivalently:
This sequence consists of the numbers A007954(k) such that A031346(k) = 3,
These are the numbers k in A002473 such that A031346(k) = 2,
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 2 steps.
Postulated to be finite and complete.
The largest known number is 2^25 * 3^227 * 7^28 (140 digits).
No more numbers have been found between 10^140 and probably 10^20000 (according to comment in A003001), and independently verified up to 10^10000.

			25 is in this sequence because:
- 25 goes to a single digit in 2 steps: p(25) = 10, p(10) = 0.
- 25 has ancestors 55, 155, etc. p(55) = 25.
27 is in this sequence because:
- 27 goes to a single digit in 2 steps: p(27) = 14, p(14) = 4.
- 27 has ancestors 39, 93, 333, 139, etc. p(39) = 27.

Daniel Mondot, Table of n, a(n) for n = 1..11994

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046511 (all numbers with mp of 2).

Cf. A350180, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 1, and 3 to 10 that are themselves 7-smooth numbers).

Select[Range@1400,AllTrue[First/@FactorInteger@#,#<10&]&&Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==2&] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

from math import prod
from sympy import factorint
def pd(n): return prod(map(int, str(n)))
def ok(n):
    if n <= 9 or max(factorint(n)) > 9: return False
    return (p := pd(n)) > 9 and pd(p) < 10
print([k for k in range(1400) if ok(k)]) # Michael S. Branicky, Jan 16 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).

A350183 Numbers of multiplicative persistence 4 which are themselves the product of digits of a number.

Original entry on oeis.org

378, 384, 686, 768, 1575, 1764, 2646, 4374, 6144, 6174, 6272, 7168, 8232, 8748, 16128, 21168, 23328, 27216, 28672, 32928, 34992, 49392, 59535, 67228, 77175, 96768, 112896, 139968, 148176, 163296, 214326, 236196, 393216, 642978, 691488, 774144, 777924

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 lists all numbers A007954(k) such that A031346(k) = 5.
- These are the numbers k in A002473 such that A031346(k) = 4.
Or:
- These numbers factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 4 steps.
Postulated to be finite and complete.

			384 is in this sequence because:
- 384 goes to a single digit in 4 steps: p(384)=96, p(96)=54, p(54)=20, p(20)=0.
- p(886)=384, p(6248)=384, p(18816)=384, etc.
378 is in this sequence because:
- 378 goes to a single digits in 4 steps: p(378)=168, p(168)=48, p(48)=32, p(32)=6.
- p(679)=378, p(2397)=378, p(12379)=378, etc.

Daniel Mondot, Table of n, a(n) for n = 1..142
Eric Weisstein's World of Mathematics, Multiplicative Persistence
Daniel Mondot, Multiplicative Persistence Tree

Cf. A002473 (7-smooth), A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046513 (all numbers with mp of 4).

Cf. A350180, A350181, A350182, A350184, A350185, A350186, A350187 (numbers with mp 1 to 3 and 5 to 10 that are themselves 7-smooth numbers).

mx=10^6;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)]}]; (* from A002473 *)
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==4&] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

pd(n) = if (n, vecprod(digits(n)), 0); \\ A007954
mp(n) = my(k=n, i=0); while(#Str(k) > 1, k=pd(k); i++); i; \\ A031346
isok(k) = (mp(k)==4) && (vecmax(factor(k)[,1]) <= 7); \\ Michel Marcus, Jan 25 2022

from math import prod
from sympy import factorint
def pd(n): return prod(map(int, str(n)))
def ok(n):
    if n <= 9 or max(factorint(n)) > 9: return False
    return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and pd(r) < 10
print([k for k in range(778000) if ok(k)])

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)])

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A028843 Numbers whose iterated product of digits is a prime.

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A133048 Powerback(n): reverse the decimal expansion of n, drop any leading zeros, then apply the powertrain map of A133500 to the resulting number.

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A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

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A064870 The minimal number which has multiplicative persistence 6 in base n.

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A064872 The minimal number which has multiplicative persistence 8 in base n.

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A070562 Fecundity of n.

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PARI

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A350181 Numbers of multiplicative persistence 2 which are themselves the product of digits of a number.

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Mathematica

Python

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

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