A361337 Numbers that reach 0 after a suitable series of split-and-multiply operations (see Comments for precise definition).

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

Michael S. Branicky, Table of n, a(n) for n = 1..10000

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

Cf. A031346, A034048, A361978.

Supersequence of A011540.

Sequence in context: A104801 A364325 A144140 * A034048 A199978 A342142

Adjacent sequences: A361334 A361335 A361336 * A361338 A361339 A361340

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