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A114135 Primitive numbers n such that the sums of the digits of n, n^2 and n^3 coincide (cf. A111434).

%I A114135 #24 Jul 07 2020 13:59:06
%S A114135 1,468,585,5851,5868,28845,58968,21688965,29588877,37848897,49879981,
%T A114135 58577797,79898994,79958368,79979698,89757468,109699677,159699969,
%U A114135 468957888,479597652,479896587,480749985,494899398,497349981,498678256
%N A114135 Primitive numbers n such that the sums of the digits of n, n^2 and n^3 coincide (cf. A111434).
%C A114135 Members of A111434 not congruent to 0 (mod 10). If k is a member of A111434 then so is 10^e*k.
%C A114135 The authors have calculated all members below 10^11.
%C A114135 The number of members less than 10^n {n=0..11}: 0,1,1,3,5,7,7,7,16,34,57,125.
%C A114135 Number of members congruent to k (mod 10): 0,7,1,0,2,23,8,20,49,15. But more interesting, number of members are congruent to k (mod 9): 66,59,0,0,0,0,0,0,0.
%C A114135 A007953(n) == n mod 9.  Since 0 and 1 are the only k in [0,1,...8] with k == k^2 mod 9, all terms are congruent to 0 or 1 mod 9. - _Robert Israel_, Jan 26 2015
%H A114135 Toshitaka Suzuki and Nikhil Mahajan, <a href="/A114135/b114135.txt">Table of n, a(n) for n = 1..600</a> (first 325 terms from Toshitaka Suzuki)
%t A114135 sod[n_] := Plus @@ IntegerDigits@n; lst = {}; Do[ If[(Mod[n, 9] == 0 || Mod[n, 9] == 1) && Mod[n, 10] != 0 && sod@n == sod[n2] == sod[n3], AppendTo[lst, n]], {n, 108/2}]; lst
%t A114135 Select[Range[5*10^8],Length[Union[Total/@IntegerDigits/@{#,#^2,#^3}]]==1 && Mod[#,10]!=0&] (* _Harvey P. Dale_, Jul 07 2020 *)
%o A114135 (PARI) isok(n) = (n % 10) && ((sd=sumdigits(n)) == sumdigits(n^2)) && (sd == sumdigits(n^3)); \\ _Michel Marcus_, Jan 20 2015
%Y A114135 Cf. A007953, A111434, A005188.
%K A114135 base,nonn
%O A114135 1,2
%A A114135 _Giovanni Resta_ and _Robert G. Wilson v_, Nov 21 2005