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%I #25 Sep 08 2022 08:46:26
%S 26,246,254,256,264,776,784,786,794,796,804,806,824,826,834,836,2454,
%T 2456,2464,2466,2474,2476,2484,2486,2494,2496,2504,2506,2514,2516,
%U 2524,2526,2534,2536,2544,2546,2554,2556,2564,2566,2594,2596,2604,2606,2614,2616,2624,2626,2634,2636,2644,7746
%N Positive numbers whose square starts and ends with exactly one 6.
%C When a square ends with 6, it ends with only one 6.
%C From _Marius A. Burtea_, Oct 30 2021 : (Start)
%C The sequence is infinite because the numbers 806, 8006, 80006, ..., 8*10^k + 6, k >= 2, are terms with squares 649636, 64096036, 6400960036, 640009600036, ..., 64*10^(2*k) + 96*10^k + 36, k >= 2.
%C Numbers 796, 7996, 79996, 799996, 7999996, 79999996, ..., 10^k*8 - 4, k >= 2, are terms and have no digits 0, because their squares are 633616, 63936016, 6399360016, 639993600016, 63999936000016, 6399999360000016, ....
%C Also 794, 7994, 79994, 799994, ..., (8*10^k - 6), k >= 2, are terms and have no digits 0, because their squares are 630436, 63904036, 6399040036, 639990400036, 63999904000036, 6399999040000036, ... (End)
%e 26^2 = 676, hence 26 is a term.
%e 814^2 = 662596, hence 814 is not a term.
%t Select[Range[10, 7750], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 6 && d[[2]] != 6 &] (* _Amiram Eldar_, Oct 30 2021 *)
%o (Python)
%o from itertools import count, takewhile
%o def ok(n):
%o s = str(n*n); return len(s.rstrip("6")) == len(s.lstrip("6")) == len(s)-1
%o def aupto(N):
%o r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [4, 6]))
%o return [k for k in r if ok(k)]
%o print(aupto(2644)) # _Michael S. Branicky_, Oct 29 2021
%o (PARI) isok(k) = my(d=digits(sqr(k))); (d[1]==6) && (d[#d]==6) && if (#d>2, (d[2]!=6) && (d[#d-1]!=6), 1); \\ _Michel Marcus_, Oct 30 2021
%o (Magma) [n:n in [4..7500]|Intseq(n*n)[1] eq 6 and Intseq(n*n)[#Intseq(n*n)] eq 6 and Intseq(n*n)[-1+#Intseq(n*n)] ne 6 ]; // _Marius A. Burtea_, Oct 30 2021
%Y Cf. A045789, A045860, A273373 (squares ending with 6).
%Y Similar to: A348487 (k=1), A348488 (k=4), A348489 (k=5), this sequence (k=6), A348491 (k=9).
%Y Subsequence of A305719.
%K nonn,base
%O 1,1
%A _Bernard Schott_, Oct 29 2021
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