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

Author

N. J. A. Sloane, Apr 01 2023, based on a posting to the Sequence Fans mailing list by Eric Angelini, Mar 20 2023

Keywords

• nonn
• base

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

Examples

			We see that 115 reaches 0 when split into 11*5: 11*5 = 55 -> 5*5 = 25 -> 2*5 = 10 -> 1*0 = 0.
		

Links

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

Crossrefs

Cf. A031346, A034048, A361978.

Supersequence of A011540.

Programs

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

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

Formula

a(2894 + k) = 3112 + k for all k >= 0 (conjectured). - M. F. Hasler, Apr 05 2023

Extensions

a(38) and beyond from Michael S. Branicky, Apr 02 2023