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A361337
Numbers that reach 0 after a suitable series of split-and-multiply operations (see Comments for precise definition).
12
0, 10, 20, 25, 30, 40, 45, 50, 52, 54, 55, 56, 58, 59, 60, 65, 69, 70, 78, 80, 85, 87, 90, 95, 96, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 115, 120, 125, 128, 129, 130, 134, 135, 136, 138, 140, 144, 145, 150, 152, 153, 154, 155, 156, 157, 158, 159
OFFSET
1,2
COMMENTS
We always split the integer N into two integers, then multiply them (and iterate). For example, 2023 can be split into 20 and 23 (producing 20*23 = 460), or split into 202 and 3 (producing 202*3 = 606). The split 2 and 023 is forbidden, as 023 is not an integer (but 460 can be split into 46 and 0 as 0 is an integer).
The sequence lists numbers which reach 0 after a suitable sequence of splits and multiplications.
If we multiply ALL the digits at each step, we get A034048 (115 is the first term where they differ).
The complement (A361978) appears to be finite, containing only 219 members, the largest being 3111. - Michael S. Branicky, Apr 02 2023
More precisely, {811, 911, 913, 921, 1111, 1112, 1113, 1121, 1122, 1131, 1211, 1231, 1261, 1311, 1321, 1612, 2111, 2121, 2211, 3111} are the only numbers not in the sequence, between 792 and at least 10^7. - M. F. Hasler, Apr 05 2023
LINKS
FORMULA
a(2894 + k) = 3112 + k for all k >= 0 (conjectured). - M. F. Hasler, Apr 05 2023
EXAMPLE
We see that 115 reaches 0 when split into 11*5: 11*5 = 55 -> 5*5 = 25 -> 2*5 = 10 -> 1*0 = 0.
PROG
(Python)
def ok(n):
if n < 10: return n == 0
s = str(n)
if "0" in s: return True
return any(ok(int(s[:i])*int(s[i:])) for i in range(1, len(s)))
print([k for k in range(116) if ok(k)]) # Michael S. Branicky, Apr 02 2023
(Python) ok = lambda n: '0' in (s:=str(n)) or any(ok(int(s[:i])*int(s[i:])) for i in range(1, len(s))) # M. F. Hasler, Apr 05 2023
(PARI) select( {is_A361337(n)=!vecmin(digits(n))|| for(p=1, logint(n, 10), is_A361337(vecprod(divrem(n, 10^p)))&& return(1))}, [1..160]) \\ M. F. Hasler, Apr 05 2023
CROSSREFS
Supersequence of A011540.
Sequence in context: A104801 A364325 A144140 * A034048 A199978 A342142
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Apr 01 2023, based on a posting to the Sequence Fans mailing list by Eric Angelini, Mar 20 2023
EXTENSIONS
a(38) and beyond from Michael S. Branicky, Apr 02 2023
STATUS
approved