A118711 - OEIS

%I #19 Aug 02 2024 08:57:23

%S 1,13,61,82,898,2962,2989,9133,20077,20653,28669,29266,35581,35842,

%T 37501,99133,236674,286717,424621,424957,821698,941650,1704301,

%U 1722370,2978413,3328258,4494466,10022317,40392829,49870141,50668882,53933053

%N Integers k such that the k-th triangular number t_k has all its base-12 digits contained in {1,5,7,11}.

%C In base 12 all primes greater than 3 end in the digits 1, 5, 7, E, where X is 10 and E is 11. They are the digits that satisfy GCD(d,12)=1.

%C The sequence in base 12 is: 1, 11, 51, 6X, 62X, 186X, 1891, 5351, E751, EE51, 14711, 14E2X, 18711, 188XX, 19851, 49451, E4E6X, 119E11, 185891, 185E11, 33762X, 394E2X, 6X2351, 6E08XX, EE7751, 11460XX, 1608E6X, 3433E51, 1163E591, 14850051, 14E7632X, 1608E311, 18331451, 1870E191, 1974E311, ..., . Note that all elements end in 1 or X. The corresponding triangular numbers after the first end in the digits 17 or 77, but not respectively.

%F k is a term if the k-th triangular number t_k = k*(k+1)/2 has its base-12 digits contained in {1,5,7,11}.

%F A000217(a(n)) = A118706(n), or equivalently, a(n) = (sqrt(8*A118706(n)+1)-1)/2. - _Amiram Eldar_, Aug 02 2024

%e 82 = 6X_12 is a term since the triangular number t=82*(82+1)/2 = 3403 = 1E77_12.

%p L:=[]: pd:={1,5,7,11}: for w to 1 do for n from 1 to 10^6 do t:=n*(n+1)/2; lod:=convert(t,base,12); sod:=convert(lod,set); if sod subset pd then L:=[op(L), [n,t]] fi; od od; L;

%t fQ[n_] := Union@ Join[{1, 5, 7, 11}, IntegerDigits[n(n + 1)/2, 12]] == {1, 5, 7, 11}; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]], {n, 10^8}] (* _Robert G. Wilson v_ *)

%Y Cf. A000217, A118706, A119033, A119034.

%K nonn,base

%O 1,2

%A _Walter Kehowski_, May 24 2006

%E Edited and extended (a(23)-a(32)) by _Robert G. Wilson v_, Jun 20 2006