A376303 - OEIS

%I #11 Sep 22 2024 18:00:51

%S 11,13,17,19,31,33,37,39,51,53,55,57,59,65,71,73,77,79,85,91,93,95,97,

%T 99,102,106,110,111,113,114,117,118,119,130,131,133,137,139,142,146,

%U 151,153,154,155,157,158,159,165,170,171,173,177,179,182,185,186,190,191,193,194,195,197,198,199

%N Numbers k such that no 7-smooth number ends in k.

%C Numbers k such that A376301(k) = -1.

%C Numbers k such that every number ending in k is divisible by a prime >= 11.

%C If i*k is a term where 2 <= i <= 9 and i*k has the same number of digits as k, then k is a term.

%H Robert Israel, <a href="/A376303/b376303.txt">Table of n, a(n) for n = 1..10000</a>

%e The multiplicative order of 3 mod 100 is 20, and that of 7 is 4.  Of the 36 numbers from 10 to 99 that are coprime to 10, the only ones congruent to some 3^i * 7^j (mod 100) are those where i = 0 to 19 and j = 0 to 3: 21, 23, 27, 29, 41, 43, 47, 49, 61, 63, 67, 69, 81, 83, 87, and 89.  The remaining 20: 11, 13, 17, 19, 31, 33, 37, 39, 51, 53, 57, 59, 71, 73, 77, 79, 91, 93, 97, 99, are members of this sequence.

%p dmax:= 3:  # for terms < 10^dmax

%p N:= 10^dmax-1: R:= NULL:

%p with(priqueue):

%p V[0]:= 10:

%p for d from 1 to dmax do

%p   A:= Array(0..10^d-1, -1);

%p   initialize(pq);

%p   insert([-1, 0, 0, 0, 0], pq);

%p   while pq[0] <> 0 do

%p     t:= extract(pq);

%p     x:= -t[1];

%p     xd:= x mod 10^d;

%p     if A[xd] = -1 then

%p       A[xd]:= x;

%p       insert([-7*x, t[2], t[3], t[4], t[5]+1], pq);

%p       if t[5] = 0 then

%p         insert([-5*x, t[2], t[3], t[4]+1, 0], pq);

%p       if t[4] = 0 then

%p         insert([-3*x, t[2], t[3]+1, 0, 0], pq);

%p       if t[3] = 0 then

%p         insert([-2*x, t[2]+1, 0, 0, 0], pq);

%p   fi fi fi fi od;

%p   R:= R, op(select(t -> A[t] = -1, {$10^(d-1)..10^d-1}))):

%p od:

%p R;

%Y Cf. A002473, A376301.

%K nonn,base

%O 1,1

%A _Robert Israel_, Sep 19 2024