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%I #32 Sep 08 2022 08:46:26
%S 212,2112,6638,6662,6688,20988,21012,21062,21112,21138,21162,21188,
%T 21212,66338,66362,66388,66412,66438,66488,66512,66562,66588,66612,
%U 66712,66738,66762,66788,66812,66838,66862,66888,66912,66938,66988,67012,67062,209762,209788
%N Positive numbers whose square starts and ends with exactly 44, and no 444.
%C When a square starts and ends with digits dd, then dd is necessarily 44.
%C The last 2 digits of terms are either 12, 38, 62 or 88.
%C From _Marius A. Burtea_, Nov 09 2021 : (Start)
%C The sequence is infinite because the numbers 212, 2112, 21112, ..., (19*10^k + 8) / 9, k >= 3, are terms because the remainder when dividing by 1000 is 544 and 445*10^(2*k - 2) < ((19*10^k + 8) / 9)^2 < 447*10^(2*k - 2), k >= 3.
%C Also 6638, 66338, 663338, 6633338, 66333338, 663333338, 6633333338, ..., (199*10^k + 14) / 3, k >= 2, are terms and have no digits 0, because their squares are: 44063044, 4400730244, 4400730244, 440017302244, 44001173022244, 4400111730222244, 440011117302222244, ... (End)
%e 212 is a term since 212^2 = 44944.
%e 662 is not a term since 662^2 = 438244.
%e 668 is not a term since 668^2 = 446224.
%e 2108 is not a term since 2108^2 = 4443664.
%e 21038 is not a term since 21038^2 = 442597444.
%e 21088 is not a term since 21088^2 = 444703744.
%t Select[Range[10, 300000], (d = IntegerDigits[#^2])[[1 ;; 2]] == d[[-2 ;; -1]] == {4, 4} && d[[-3]] != 4 && d[[3]] != 4 &] (* _Amiram Eldar_, Nov 08 2021 *)
%o (Magma) fd:=func<n|Seqint(Intseq(n*n)) mod 100 eq 44 and Seqint(Intseq(n*n)) mod 1000 ne 444>; fs:=func<n|Seqint(Reverse(Intseq(n*n))) mod 100 eq 44 and Seqint(Reverse(Intseq(n*n))) mod 1000 ne 444>; [n:n in [1..210000]|fd(n) and fs(n)]; // _Marius A. Burtea_, Nov 08 2021
%o (Python)
%o from itertools import count, takewhile
%o def ok(n):
%o s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-2
%o def aupto(N):
%o ends = [12, 38, 62, 88]
%o r = takewhile(lambda x: x<=N, (100*i+d for i in count(0) for d in ends))
%o return [k for k in r if ok(k)]
%o print(aupto(209788)) # _Michael S. Branicky_, Nov 08 2021
%Y Cf. A017317.
%Y Subsequence of A045858, A273375, A305719 and A346774.
%Y Similar to: A348488 (d=4), this sequence (dd=44), A348832 (ddd=444).
%K nonn,base
%O 1,1
%A _Bernard Schott_, Nov 08 2021
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