%I #29 Sep 08 2022 08:46:26

%S 2,22,68,202,208,218,222,642,648,652,658,672,678,682,692,698,702,2002,

%T 2008,2018,2022,2028,2032,2042,2048,2052,2058,2068,2072,2078,2082,

%U 2092,2122,2128,2132,2142,2148,2152,2158,2168,2172,2178,2182,2192,2198,2202,2208,2218,2222,2228

%N Positive numbers whose square starts and ends with exactly one 4.

%C When a square ends with 4 (A273375), this square may end with precisely one 4, two 4's or three 4's (A328886).

%C This sequence is infinite as each 2*(10^m + 1), m >= 1 or 2*(10^m + 4), m >= 2 is a term.

%C Numbers 2, 22, 222, ..., 2*(10^k - 1) / 9, (k >= 1), as well as numbers 2228, 22228, ..., 2*(10^k + 52) / 9, (k >= 4) are terms and have no digits 0. - _Marius A. Burtea_, Oct 24 2021

%e 22 is a term since 22^2 = 484.

%e 638 is not a term since 638^2 = 407044.

%e 668 is not a term since 668^2 = 446224.

%t Join[{2}, Select[Range[10, 2000], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 4 && d[[-2]] != 4 && d[[2]] != 4 &]] (* _Amiram Eldar_, Oct 24 2021 *)

%o (PARI) isok(k) = my(d=digits(sqr(k))); (d[1]==4) && (d[#d]==4) && if (#d>2, (d[2]!=4) && (d[#d-1]!=4), 1); \\ _Michel Marcus_, Oct 24 2021

%o (Magma) [2] cat [n:n in [4..2300]|Intseq(n*n)[1] eq 4 and Intseq(n*n)[#Intseq(n*n)] eq 4 and Intseq(n*n)[-1+#Intseq(n*n)] ne 4 and Intseq(n*n)[2] ne 4]; // _Marius A. Burtea_, Oct 24 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)-1

%o def aupto(N):

%o r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [2, 8]))

%o return [k for k in r if ok(k)]

%o print(aupto(2228)) # _Michael S. Branicky_, Oct 24 2021

%Y Cf. A045858, A273375 (squares ending with 4), A017317, A328886 (squares ending with three 4).

%Y Cf. A002276 \ {0} (a subsequence).

%Y Subsequence of A305719.

%Y Similar to: A348487 (k=1), this sequence (k=4), A348489 (k=5), A348490 (k=6).

%K nonn,base

%O 1,1

%A _Bernard Schott_, Oct 24 2021