A350187 -id:A350187 - cp's OEIS frontend

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

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

A238985 Zeroless 7-smooth numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 32, 35, 36, 42, 45, 48, 49, 54, 56, 63, 64, 72, 75, 81, 84, 96, 98, 112, 125, 126, 128, 135, 144, 147, 162, 168, 175, 189, 192, 196, 216, 224, 225, 243, 245, 252, 256, 288, 294, 315, 324, 336

Offset: 1

Charles R Greathouse IV and Reinhard Zumkeller, Mar 07 2014

A001221(a(n)) <= 3 since 10 cannot divide a(n).

It seems that this sequence is finite and contains 12615 terms. - Daniel Mondot, May 03 2022 and Jianing Song, Jan 28 2023

			a(12615) = 2^25 * 3^227 * 7^28.

Daniel Mondot, Table of n, a(n) for n = 1..12615 (terms 1..10000 from Charles R Greathouse IV)

Cf. A168046, intersection of A002473 and A052382.
A238938, A238939, A238940, A195948, A238936, A195908 are proper subsequences.
Cf. A059405 (subsequence), A350180 through A350187.

import Data.Set (singleton, deleteFindMin, fromList, union)
a238985 n = a238985_list !! (n-1)
a238985_list = filter ((== 1) . a168046) $ f $ singleton 1 where
   f s = x : f (s' `union` fromList
               (filter ((> 0) . (`mod` 10)) $ map (* x) [2,3,5,7]))
               where (x, s') = deleteFindMin s

zf(n)=vecmin(digits(n))
list(lim)=my(v=List(),t,t1); for(e=0,log(lim+1)\log(7), t1=7^e; for(f=0,log(lim\t1+1)\log(3), t=t1*3^f; while(t<=lim, if(zf(t), listput(v, t)); t<<=1)); for(f=0,log(lim\t1+1)\log(5), t=t1*5^f; while(t<=lim, if(zf(t), listput(v, t)); t*=3))); Set(v)

A086299(a(n)) * A168046(a(n)) = 1.

Keyword:fini and keyword:full removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured.

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)])

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)])

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 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A238985 Zeroless 7-smooth numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

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

Views

Author