A033180 -id:A033180 - cp's OEIS frontend

A067109 Number of occurrences of the string n in n! (A000142).

1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 2, 4, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 2, 5, 0

Offset: 1

Amarnath Murthy, Jan 08 2002

a(A033180(n)) > 0. - Reinhard Zumkeller, Aug 23 2008

			a(4) = 1 as 4! = 24 and 4 occurs once;
a(5) = 0 as 5! = 120 does not contain a 5;
a(20) = 1 as 20! = 2432902008176640000 and 20 occurs once.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000142, A033180.

import Data.List (tails, isPrefixOf)
a067109 n = sum $
   map (fromEnum . (show n `isPrefixOf`)) (tails $ show $ a000142 n)
-- Reinhard Zumkeller, Aug 28 2014

Table[ Length[ StringPosition[ ToString[n! ], ToString[n]]], {n, 1, 75} ]
Table[SequenceCount[IntegerDigits[n!],IntegerDigits[n],Overlaps->True],{n,100}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 01 2019 *)

More terms from Robert G. Wilson v, Jan 09 2002

A383217 Lexicographically earliest strictly increasing sequence such that no term is a substring of the product of all previous terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 40, 41, 44, 45, 46, 48, 49, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 76, 79, 80, 84, 85, 87, 90, 91, 97, 98

Offset: 1

Dominic McCarty, Apr 19 2025

			The product of the first 6 terms is 720. "7" is a substring of "720", so a(7) cannot be 7. So, a(7) is the next available value, 8.

Dominic McCarty, Table of n, a(n) for n = 1..10000

Cf. A383218 (product of first n terms), A033180.

from itertools import count
from math import prod
a = [1]
while len(a) < 40: a.append(next(k for k in count(a[-1]+1) if str(k) not in str(prod(a))))
print(a)

A383218 The product of the first n terms of A383217.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5760, 51840, 518400, 5702400, 68428800, 889574400, 12454041600, 186810624000, 2988969984000, 50812489728000, 914624815104000, 17377871486976000, 347557429739520000, 7298706024529920000, 160571532539658240000, 3693145248412139520000

Offset: 1

Dominic McCarty, Apr 19 2025

Dominic McCarty, Table of n, a(n) for n = 1..429

Cf. A383217, A033180.

from itertools import count
a, p = [1], 1
for k in count(2):
    if str(k) not in str(p): p *= k; a.append(p)
    if len(a) >= 20: break
print(a)

A113621 Numbers k such that the representation of k^2 is a substring of that of k!, in base 10.

Original entry on oeis.org

1, 20, 29, 170, 176, 241, 3136, 9800, 20309, 20486, 53663, 73793, 94836, 200000

Offset: 1

Giovanni Resta, Jan 26 2006

Using one of the fast algorithms for computing the last nonzero digit of the factorial (A008904) it is easy to see that also 200000000 and 2*10^16 are terms.

			29^2 = 841 and 29! = 8(841)761993739701954543616000000.

Cf. A033180, A008904.

CF. A000142, A000290.

lst={}; Do[If[{}!= StringPosition[ToString[n! ], ToString[n^2]], AppendTo[lst, n]], {n, 10000}]; lst

Extended by Giovanni Resta, Apr 04 2014

cp's OEIS Frontend

A067109 Number of occurrences of the string n in n! (A000142).

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A383217 Lexicographically earliest strictly increasing sequence such that no term is a substring of the product of all previous terms.

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Python

A383218 The product of the first n terms of A383217.

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Author

Keywords

Links

Crossrefs

Programs

Python

A113621 Numbers k such that the representation of k^2 is a substring of that of k!, in base 10.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions