%I #36 Sep 08 2022 08:46:26

%S 75,225,715,725,735,755,765,2245,2255,2265,2275,2285,2295,2305,2315,

%T 2325,2335,2345,2375,2385,2395,2405,2415,2425,2435,2445,7075,7085,

%U 7095,7105,7115,7125,7135,7145,7155,7165,7175,7185,7195,7205,7215,7225,7235,7245

%N Positive numbers whose square starts and ends with exactly one 5.

%C When a square ends with 5, it ends with 25.

%C From _Marius A. Burtea_, Oct 25 2021: (Start)

%C Numbers 75, 765, 7665, 76665, ..., (23*10^k -5) / 3, k >= 1, are terms and have no digits 0, because their squares are 5625, 585225, 58752225, 5877522225, 587775222225, 58777752222225, ...

%C Also 75, 735, 7335, 73335, ..., (22*10^n+5) / 3, k >= 1, are terms and have no digits 0, because their squares are 5625, 540225, 53802225, 5378022225, 537780222225, 53777802222225, ... (End)

%e 75^2 = 5625, hence 75 is a term.

%e 235^2 = 55225, hence 235 is not a term.

%t Select[5 * Range[2, 1500], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 5 && d[[2]] != 5 &] (* _Amiram Eldar_, Oct 25 2021 *)

%o (PARI) isok(k) = my(d=digits(sqr(k))); (d[1]==5) && (d[#d]==5) && if (#d>2, (d[2]!=5) && (d[#d-1]!=5), 1); \\ _Michel Marcus_, Oct 25 2021

%o (Magma) [n:n in [4..7500]|Intseq(n*n)[1] eq 5 and Intseq(n*n)[#Intseq(n*n)] eq 5 and Intseq(n*n)[-1+#Intseq(n*n)] ne 5 ]; // _Marius A. Burtea_, Oct 25 2021

%o (Python)

%o from itertools import count, takewhile

%o def ok(n):

%o s = str(n*n); return len(s.rstrip("5")) == len(s.lstrip("5")) == len(s)-1

%o def aupto(N):

%o r = takewhile(lambda x: x<=N, (10*i+5 for i in count(0)))

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

%o print(aupto(7245)) # _Michael S. Branicky_, Oct 26 2021

%Y Cf. A045859, A017330 (squares ending with 5).

%Y Similar to: A348487 (k=1), A348488 (k=4), this sequence (k=5), A348490 (k=6), A348491 (k=9).

%Y Subsequence of A305719.

%K nonn,base

%O 1,1

%A _Bernard Schott_, Oct 25 2021