A365994 - OEIS

%I #15 Sep 27 2023 13:50:28

%S 101,3981053,103,1196913103,4185108867,2053,107,13093,109,50821,

%T 1319023,206,1013,2309,1970359,1160555267,1435393103,5147059,1019,

%U 24786109,307,1265393034541,11093,1104119,505,11409,309,44661214906829,191077,1210977611,1031,18107519,1033,33269991281067,170767

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

%C 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).

%H Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2023/09/multiply-with-zero.html">Multiply with zero</a>, personal blog "Cinquante signes" Sept. 2023.

%e Trajectories of n for n = 1 to 40.

%e A "stop" means that the next term will contain two or more zeros.

%e n = 1 --> 1, 101, stop

%e n = 2 --> 2, 102, 1706, 20853, 210993, 3981053, stop

%e n = 3 --> 3, 103, stop

%e n = 4 --> 4, 104, 1308, 20654, 230898, 2654087, 35907393, 1196913103, stop

%e n = 5 --> 5, 105, 1507, 110137, 2410457, 34435107, 371092817, 4185108867, stop

%e n = 6 --> 6, 106, 2053, stop

%e n = 7 --> 7, 107, stop

%e n = 8 --> 8, 108, 1209, 13093, stop

%e n = 9 --> 9, 109, stop

%e n = 10 --> 10, 205, 4105, 50821, stop

%e n = 11 --> 11, 1011, 30337, 1319023, stop

%e n = 12 --> 12, 206, stop

%e n = 13 --> 13, 1013, stop

%e n = 14 --> 14, 207, 2309, stop

%e n = 15 --> 15, 305, 5061, 70723, 1970359, stop

%e n = 16 --> 16, 208, 2608, 32608, 408152, 6260652, 64410972, 1160555267, stop

%e n = 17 --> 17, 1017, 11309, 263043, 2657099, 43061793, 1435393103, stop

%e n = 18 --> 18, 209, 11019, 303673, 5147059, stop

%e n = 19 --> 19, 1019, stop

%e n = 20 --> 20, 405, 4509, 167027, 2230749, 24786109, stop

%e n = 21 --> 21, 307, stop

%e n = 22 --> 22, 1022, 14073, 304691, 17017923, 305672641, 3469610881, 43707939613, 1265393034541, stop

%e n = 23 --> 23, 1023, 11093, stop

%e n = 24 --> 24, 308, 4077, 45309, 1104119, stop

%e n = 25 --> 25, 505, stop

%e n = 26 --> 26, 1026, 11409, stop

%e n = 27 --> 27, 309, stop

%e n = 28 --> 28, 407, 11037, 130849, 1707697, 17770961, 1301366997, 14459633309, 149743096563, 3049914365521, 44661214906829, stop

%e n = 29 --> 29, 1029, 14707, 191077, stop

%e n = 30 --> 30, 506, 11046, 140789, 11012799, 118290931, 1210977611, stop

%e n = 31 --> 31, 1031, stop

%e n = 32 --> 32, 408, 5108, 127704, 1360939, 18107519, stop

%e n = 33 --> 33, 1033, stop

%e n = 34 --> 34, 1034, 11094, 129086, 1590817, 21075277, 253919083, 15701617319, 199307878383, 2229089415827, 33269991281067, stop

%e n = 35 --> 35, 507, 13039, 170767, stop

%e n = 36 --> 36, 409, stop

%e n = 37 --> 37, 1037, 17061, 305687, stop

%e n = 38 --> 38, 1038, 17306, 208653, 3069551, stop

%e n = 39 --> 39, 1039, stop

%e n = 40 --> 40, 508, 12704, 158808, 1985108, 20992554, 306997518, 5116625306, 13032065196793281, stop

%e etc.

%t Table[Last@Most@NestWhileList[(d=Divisors@#;

%t 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}]

%o (Python)

%o from sympy import divisors

%o def a(n):

%o     k = n

%o     while True:

%o         s = set()

%o         for d in divisors(k, generator=True):

%o             v, w = str(d), str(k//d)

%o             if "0" not in v and "0" not in w:

%o                 s.add(int(v + "0" + w))

%o         if len(s) == 0: return k

%o         k = min(s)

%o print([a(n) for n in range(1, 41)]) # _Michael S. Branicky_, Sep 26 2023

%Y Cf. A365993, A365260.

%K base,nonn

%O 1,1

%A _Eric Angelini_ and _Giorgos Kalogeropoulos_, Sep 25 2023