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A061808 a(n) is the smallest number with all digits odd that is divisible by 2n-1.

%I A061808 #57 Mar 06 2025 11:06:16
%S A061808 1,3,5,7,9,11,13,15,17,19,315,115,75,135,319,31,33,35,37,39,533,559,
%T A061808 135,517,539,51,53,55,57,59,793,315,195,335,759,71,73,75,77,79,1377,
%U A061808 913,595,957,979,91,93,95,97,99,1111,515,315,535,1199,111,113,115,117,119,1331,1353,375,1397,1935
%N A061808 a(n) is the smallest number with all digits odd that is divisible by 2n-1.
%C A061808 From _Yang Haoran_, Dec 02 2017, edited by _M. F. Hasler_, Mar 05 2025: (Start)
%C A061808 Record value for a(n) = (2n-1) * A296009(n):
%C A061808    (1, 3, 5, ..., 19) * 1 = (1, 3, 5, ..., 19)
%C A061808    21 *  15 =    315
%C A061808    29 *  11 =    319
%C A061808    41 *  13 =    533
%C A061808    43 *  13 =    559
%C A061808    61 *  13 =    793
%C A061808    81 *  17 =   1377
%C A061808   127 *  11 =   1397
%C A061808   129 *  15 =   1935
%C A061808   149 *  13 =   1937
%C A061808   167 *  19 =   3173
%C A061808   169 *  33 =   5577
%C A061808   201 * 155 =  31155
%C A061808   299 * 105 =  31395
%C A061808   401 * 133 =  53333
%C A061808   601 * 119 =  71519
%C A061808   633 * 283 = 179139
%C A061808   (complete up to here)
%C A061808   ...
%C A061808   990001 * 12121113 = 11999913991113 (the first A296009(n) > 2n-1).
%C A061808 (End)
%C A061808 All terms must be odd. - _M. F. Hasler_, Mar 05 2025
%H A061808 Robert Israel, <a href="/A061808/b061808.txt">Table of n, a(n) for n = 1..10000</a>
%H A061808 Mathematical Excalibur, <a href="https://www.math.ust.hk/excalibur/v13_n1.pdf">Problem 300</a>, Vol. 1 No. 3 (2008), p. 3.
%F A061808 From  _M. F. Hasler_, Mar 05 2025: (Start)
%F A061808 a(n) = (2n-1)*A296009(n).
%F A061808 a(n) == 1 (mod 2) for all n. (End)
%p A061808 Ad[1]:= [1,3,5,7,9]:
%p A061808 for n from 2 to 9 do Ad[n]:= map(t -> seq(10*t+j,j=[1,3,5,7,9]), Ad[n-1]) od:
%p A061808 Aod:= [seq(op(Ad[i]),i=1..9)]:
%p A061808 f:= proc(n) local k;
%p A061808    for k from 1 to nops(Aod) do
%p A061808        if Aod[k] mod (2*n-1) = 0 then return(Aod[k]) fi
%p A061808      od;
%p A061808      NotFound
%p A061808 end proc:
%p A061808 map(f, [$1..100]); # _Robert Israel_, Feb 15 2017
%t A061808 Table[Block[{k = 2 n - 1}, While[Nand[AllTrue[IntegerDigits@ k, OddQ], Divisible[k, 2 n - 1]], k += 2]; k], {n, 59}] (* _Michael De Vlieger_, Dec 02 2017 *)
%o A061808 (Magma) a:=[]; for n in [1..120 by 2] do k:=1; while not Set(Intseq(n*k)) subset {1,3,5,7,9} do k:=k+2; end while; Append(~a,k*n); end for; a; // _Marius A. Burtea_, Sep 20 2019
%o A061808 (PARI) isoddd(n) = #select(x->((x%2) == 0), digits(n)) == 0;
%o A061808 a(n) = {my(m = 2*n-1, k = 1); while(!isoddd(k*m), k++); k*m;} \\ _Michel Marcus_, Sep 20 2019
%o A061808 (PARI) apply( {A061808(n)=forstep(k=n*2-1,oo,n*4-2,vecmin(digits(k)%2)&& return(k))}, [1..99])
%o A061808 (Python) A061808 = lambda n: next(m for m in range(2*n-1,9<<99,4*n-2) if all(int(d)&1 for d in str(m))) # _M. F. Hasler_, Mar 05 2025
%Y A061808 Cf. A061807, A014261.
%Y A061808 Equals A296009 * (2n-1).
%K A061808 base,nonn,look
%O A061808 1,2
%A A061808 _Amarnath Murthy_, May 28 2001
%E A061808 Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 30 2001