A365994 The n-th term of the sequence is the last term of n's trajectory under the "multiply with zero" rules explained in A365993.

101, 3981053, 103, 1196913103, 4185108867, 2053, 107, 13093, 109, 50821, 1319023, 206, 1013, 2309, 1970359, 1160555267, 1435393103, 5147059, 1019, 24786109, 307, 1265393034541, 11093, 1104119, 505, 11409, 309, 44661214906829, 191077, 1210977611, 1031, 18107519, 1033, 33269991281067, 170767

Offset

1,1

Comments

It is conjectured that all trajectories will stop quite rapidly, mostly because the probability for 0 to appear in a divisor of a(n) increases with the size of a(n).

Links

Table of n, a(n) for n=1..35.

Eric Angelini, Multiply with zero, personal blog "Cinquante signes" Sept. 2023.

Example

Trajectories of n for n = 1 to 40.
A "stop" means that the next term will contain two or more zeros.
n = 1 --> 1, 101, stop
n = 2 --> 2, 102, 1706, 20853, 210993, 3981053, stop
n = 3 --> 3, 103, stop
n = 4 --> 4, 104, 1308, 20654, 230898, 2654087, 35907393, 1196913103, stop
n = 5 --> 5, 105, 1507, 110137, 2410457, 34435107, 371092817, 4185108867, stop
n = 6 --> 6, 106, 2053, stop
n = 7 --> 7, 107, stop
n = 8 --> 8, 108, 1209, 13093, stop
n = 9 --> 9, 109, stop
n = 10 --> 10, 205, 4105, 50821, stop
n = 11 --> 11, 1011, 30337, 1319023, stop
n = 12 --> 12, 206, stop
n = 13 --> 13, 1013, stop
n = 14 --> 14, 207, 2309, stop
n = 15 --> 15, 305, 5061, 70723, 1970359, stop
n = 16 --> 16, 208, 2608, 32608, 408152, 6260652, 64410972, 1160555267, stop
n = 17 --> 17, 1017, 11309, 263043, 2657099, 43061793, 1435393103, stop
n = 18 --> 18, 209, 11019, 303673, 5147059, stop
n = 19 --> 19, 1019, stop
n = 20 --> 20, 405, 4509, 167027, 2230749, 24786109, stop
n = 21 --> 21, 307, stop
n = 22 --> 22, 1022, 14073, 304691, 17017923, 305672641, 3469610881, 43707939613, 1265393034541, stop
n = 23 --> 23, 1023, 11093, stop
n = 24 --> 24, 308, 4077, 45309, 1104119, stop
n = 25 --> 25, 505, stop
n = 26 --> 26, 1026, 11409, stop
n = 27 --> 27, 309, stop
n = 28 --> 28, 407, 11037, 130849, 1707697, 17770961, 1301366997, 14459633309, 149743096563, 3049914365521, 44661214906829, stop
n = 29 --> 29, 1029, 14707, 191077, stop
n = 30 --> 30, 506, 11046, 140789, 11012799, 118290931, 1210977611, stop
n = 31 --> 31, 1031, stop
n = 32 --> 32, 408, 5108, 127704, 1360939, 18107519, stop
n = 33 --> 33, 1033, stop
n = 34 --> 34, 1034, 11094, 129086, 1590817, 21075277, 253919083, 15701617319, 199307878383, 2229089415827, 33269991281067, stop
n = 35 --> 35, 507, 13039, 170767, stop
n = 36 --> 36, 409, stop
n = 37 --> 37, 1037, 17061, 305687, stop
n = 38 --> 38, 1038, 17306, 208653, 3069551, stop
n = 39 --> 39, 1039, stop
n = 40 --> 40, 508, 12704, 158808, 1985108, 20992554, 306997518, 5116625306, 13032065196793281, stop
etc.

Mathematica

Table[Last@Most@NestWhileList[(d=Divisors@#;
Min[Select[FromDigits@Flatten[IntegerDigits/@Riffle[#, 0]]&/@Table[{d[[k]], d[[Length@d-k+1]]}, {k, Length@d}], Count[IntegerDigits@#, 0]<2&]])&, n, IntegerQ], {n, 40}]

Prog

(Python)
from sympy import divisors
def a(n):
    k = n
    while True:
        s = set()
        for d in divisors(k, generator=True):
            v, w = str(d), str(k//d)
            if "0" not in v and "0" not in w:
                s.add(int(v + "0" + w))
        if len(s) == 0: return k
        k = min(s)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Sep 26 2023

Crossrefs

Cf. A365993, A365260.

Sequence in context: A138120 A266609 A082521 * A262645 A138826 A292686

Adjacent sequences: A365991 A365992 A365993 * A365995 A365996 A365997

Keyword

base,nonn

Author

Eric Angelini and Giorgos Kalogeropoulos, Sep 25 2023

Status

approved