cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A046510 Numbers with multiplicative persistence value 1.

Original entry on oeis.org

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

Views

Author

Patrick De Geest, Sep 15 1998

Keywords

Comments

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

Examples

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

Links

Crossrefs

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).

Programs

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

Extensions

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

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

Views

Author

Daniel Mondot, Jan 15 2022

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)
  • Python
    #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)])

A046506 Primes with multiplicative persistence value 6.

Original entry on oeis.org

8867, 23887, 27883, 28387, 28837, 32887, 34487, 34847, 38287, 38447, 43487, 44647, 46447, 47843, 48437, 48473, 49999, 72883, 74843, 78283, 78823, 82387, 82837, 84347, 84437, 87443, 88237, 88327, 94999, 118687, 123887, 126487, 128467
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1998

Keywords

Examples

			8867 -> [ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in six steps.

Links

Crossrefs

Cf. A046500, A046515.

Programs

  • Maple
    filter:= proc(n) local L,i;
  L:= convert(convert(n,base,10),`*`); if L < 10 then return false fi;
  for i from 2 to 5 do
     L:= convert(convert(L,base,10),`*`); if L < 10 then return false fi
    od;
  L:= convert(convert(L,base,10),`*`); evalb(L < 10)
end proc:
count:= 0: Res:= NULL;
p:= 11:
while count < 100 do
  p:= nextprime(p);
  if filter(p) then count:= count+1; Res:= Res,p fi
od:
Res; # Robert Israel, Jun 06 2018

Extensions

Offset corrected by Robert Israel, Jun 06 2018

A199996 Composite numbers whose multiplicative persistence is 6.

Original entry on oeis.org

6788, 6878, 6887, 7688, 7868, 7886, 8678, 8687, 8768, 8786, 8876, 16788, 16878, 16887, 17688, 17868, 17886, 18678, 18687, 18768, 18786, 18867, 18876, 23788, 23878, 24678, 24687, 24768, 24786, 24867, 24876, 26478, 26487, 26748, 26784, 26847, 26874, 27388
Offset: 1

Views

Author

Jaroslav Krizek, Nov 13 2011

Keywords

Comments

Complement of A046506 with respect to A046515.

Examples

			6788 -> [ 2688 ][ 768 ][ 336 ][ 54 ][ 20 ][ 0 ] -> one digit in six steps.

Crossrefs

Cf. A046506 (primes whose multiplicative persistence is 6).

Programs

  • Mathematica
    persistence[n_] := Module[{cnt = 0, k = n}, While[k > 9, cnt++; k = Times @@ IntegerDigits[k]]; cnt]; Select[Range[30000], ! PrimeQ[#] && persistence[#] == 6 &] (* T. D. Noe, Nov 23 2011 *)
Showing 1-4 of 4 results.

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