A002473 -id:A002473 - cp's OEIS frontend

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

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)])

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).

A369374 Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.

Original entry on oeis.org

36, 72, 108, 144, 216, 288, 324, 432, 576, 648, 864, 900, 972, 1152, 1296, 1728, 1800, 1944, 2304, 2592, 2700, 2916, 3456, 3600, 3888, 4500, 4608, 5184, 5400, 5832, 6912, 7200, 7776, 8100, 8748, 9000, 9216, 10368, 10800, 11664, 13500, 13824, 14400, 15552, 16200

Offset: 1

Michael De Vlieger, Jan 22 2024

nonn

Numbers k such that Omega(k) > omega(k) > 1 with all prime power factors p^m for m > 1, such that squarefree kernel rad(k) is in A002110, where Omega = A001222, omega = A001221, and rad(k) = A007947(k).
Union of the product of the squares of primorials P(n)^2, n > 1, and the set of prime(n)-smooth numbers.
Superset of A364930.
Proper subset of A367268, which in turn is a proper subset of A126706.

			This sequence is the union of the following infinite sets:
P(2)^2 * A003586 = {36, 72, 108, 144, 216, 288, 324, ...}
                 = { m*P(2)^2 : rad(m) | P(2) }.
P(3)^2 * A051037 = {900, 1800, 2700, 3600, 4500, 5400, ...}
                 = { m*P(3)^2 : rad(m) | P(3) }.
P(4)^2 * A002473 = {44100, 88200, 132300, 176400, ...}
                 = { m*P(4)^2 : rad(m) | P(4) }, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A001694, A002110, A007947, A055932, A126706, A286708, A364930, A367268, A369417.

With[{nn = 2^14},
  Select[
    Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
      Not@*PrimePowerQ],
    And[EvenQ[#],
      Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]

{a(n)} = { m*P(n)^2 : P(n) = Product_{j = 1..n} prime(n), rad(m) | P(n), n > 1 }.
Intersection of A286708 and A055932.
A286708 is the union of A369417 and this sequence.

A071604 a(n) is the number of 7-smooth numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 34, 35, 36, 36, 36, 36, 36, 36, 37, 37, 38

Offset: 1

Benoit Cloitre, Jun 02 2002

A 7-smooth number is a number of the form 2^x*3^y*5^z*7^u, (x,y,z,u) >= 0.

In other words, a 7-smooth number is a number with no prime factor greater than 7. - Peter Munn, Nov 20 2021

			a(11) = 10 as there are 10 7-smooth numbers <= 11. Namely 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - _David A. Corneth_, Apr 19 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Partial sums of A086299.
Column 7 of A080786.
Cf. A002473, A106600, A343598.
Equivalent sequences with other limits on greatest prime factor: A070939 (2), A071521 (3), A071520 (5), A071523 (11), A080684 (13), A080685 (17), A080686 (19), A096300 (log n).

for(n=1,100,print1(sum(k=1,n,if(sum(i=5,n,if(k%prime(i),0,1)),0,1)),","))

from sympy import integer_log
def A071604(n):
    c = 0
    for i in range(integer_log(n,7)[0]+1):
        i7 = 7**i
        m = n//i7
        for j in range(integer_log(m,5)[0]+1):
            j5 = 5**j
            r = m//j5
            for k in range(integer_log(r,3)[0]+1):
                c += (r//3**k).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = Card{ k | A002473 (k) <= n }.

Name corrected by David A. Corneth, Apr 19 2021

A085129 Multiples of 6 which are 7-smooth.

Original entry on oeis.org

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72, 84, 90, 96, 108, 120, 126, 144, 150, 162, 168, 180, 192, 210, 216, 240, 252, 270, 288, 294, 300, 324, 336, 360, 378, 384, 420, 432, 450, 480, 486, 504, 540, 576, 588, 600, 630, 648, 672, 720, 750, 756, 768, 810, 840

Offset: 1

Amarnath Murthy, Jul 06 2003

Equivalently, multiples of 6 with the largest prime divisor < 10.

Intersection of A008588 and A002473.

Cf. A085125, A085126, A085127, A085128, A080194, A085131, A085132.

With[{p = Prime[Range[4]]}, 6 * Select[Range[140], Times @@ (p^IntegerExponent[#, p]) == # &]] (* Amiram Eldar, Sep 23 2024 *)

From Amiram Eldar, Sep 22 2024: (Start)
a(n) = 6*A002473(n).
Sum_{n>=1} 1/a(n) = 35/48. (End)

More terms from David Wasserman, Jan 28 2005

Offset changed by Andrew Howroyd, Sep 19 2024

A085132 Multiples of 9 which are 7-smooth.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 108, 126, 135, 144, 162, 180, 189, 216, 225, 243, 252, 270, 288, 315, 324, 360, 378, 405, 432, 441, 450, 486, 504, 540, 567, 576, 630, 648, 675, 720, 729, 756, 810, 864, 882, 900, 945, 972, 1008, 1080, 1125, 1134, 1152

Offset: 1

Amarnath Murthy, Jul 06 2003

Equivalently, multiples of 9 with the largest prime divisor < 10.

Intersection of A008591 and A002473.

Cf. A085125, A085126, A085127, A085128, A085129, A080194, A085131.

With[{p = Prime[Range[4]]}, 9 * Select[Range[140], Times @@ (p^IntegerExponent[#, p]) == # &]] (* Amiram Eldar, Sep 23 2024 *)

From Amiram Eldar, Sep 22 2024: (Start)
a(n) = 9*A002473(n).
Sum_{n>=1} 1/a(n) = 35/72. (End)

More terms from David Wasserman, Jan 28 2005

Offset changed by Andrew Howroyd, Sep 19 2024

cp's OEIS Frontend

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

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

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

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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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A369374 Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.

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A071604 a(n) is the number of 7-smooth numbers <= n.

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