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%I A348491 #21 Sep 08 2022 08:46:26
%S A348491 3,97,303,307,313,953,957,963,967,973,977,983,987,993,3003,3007,3013,
%T A348491 3017,3023,3027,3033,3037,3043,3047,3053,3057,3063,3067,3073,3077,
%U A348491 3083,3087,3093,3097,3103,3107,3113,3117,3123,3127,3133,3137,3143,9487,9493,9497,9503,9507,9513,9517
%N A348491 Positive numbers whose square starts and ends with exactly one 9.
%C A348491 When a square ends with 9, it ends with only one 9.
%C A348491 From _Marius A. Burtea_, Nov 02 2021 : (Start)
%C A348491 The sequence is infinite because the numbers 303, 3003, 30003, ..., 3*(10^k + 1), k >= 2, are terms with squares 91809, 9018009, 900180009, 90001800009, ... 9*(10^(2*k) + 2*10^k + 1), k >= 2.
%C A348491 Numbers 97, 967, 9667, 96667, 966667, ..., (29*10^n + 1) / 3, k >= 1, are terms and have no digits 0, because their squares are 9409, 935089, 93450889, 9344508889, 934445088889, ...
%C A348491 Also 963, 9663, 96663, 966663, 9666663, 96666663, ... (29*10^k - 11) / 3, k >= 2, are terms and have no digits 0, because their squares are 927369, 93373569, 9343735569, 934437355569, 93444373555569, 9344443735555569, ... (End)
%e A348491 97^2 = 9409, hence 97 is a term.
%e A348491 997^2 = 994009, hence  997 is not a term.
%t A348491 Join[{3}, Select[Range[10, 10^4], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 9 && d[[2]] != 9 &]] (* _Amiram Eldar_, Nov 02 2021 *)
%o A348491 (Magma) [3] cat [n:n in [4..9600]|Intseq(n*n)[1] eq 9 and Intseq(n*n)[#Intseq(n*n)] eq 9]; // _Marius A. Burtea_, Nov 02 2021
%o A348491 (Python)
%o A348491 from itertools import count, takewhile
%o A348491 def ok(n):
%o A348491   s = str(n*n); return len(s.rstrip("9")) == len(s.lstrip("9")) == len(s)-1
%o A348491 def aupto(N):
%o A348491   r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [3, 7]))
%o A348491   return [k for k in r if ok(k)]
%o A348491 print(aupto(9517)) # _Michael S. Branicky_, Nov 02 2021
%o A348491 (PARI) isok(k) = my(d=digits(sqr(k))); (d[1]==9) && (d[#d]==9) && if (#d>2, (d[2]!=9) && (d[#d-1]!=9), 1); \\ _Michel Marcus_, Nov 03 2021
%o A348491 (PARI) list(lim)=my(v=List([3])); for(d=2, 2*#digits(lim\=1), my(s=sqrtint(9*10^(d-1)-1)+1); s+=[3,2,1,0,3,2,1,0,5,4][s%10+1]; forstep(n=s, min(sqrtint(10^d-10^(d-2)-1), lim), if(s%10==3, [4,6], [6,4]), listput(v, n))); Vec(v) \\ _Charles R Greathouse IV_, Nov 03 2021
%Y A348491 Subsequence of A305719, A063226, and A045863.
%Y A348491 Cf. A017377, A045863, A273374 (squares ending with 9).
%Y A348491 Similar to: A348487 (k=1), A348488 (k=4), A348489 (k=5), A348490 (k=6), this sequence (k=9).
%K A348491 nonn,base,easy
%O A348491 1,1
%A A348491 _Bernard Schott_, Nov 02 2021

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