A348489 Positive numbers whose square starts and ends with exactly one 5.

75, 225, 715, 725, 735, 755, 765, 2245, 2255, 2265, 2275, 2285, 2295, 2305, 2315, 2325, 2335, 2345, 2375, 2385, 2395, 2405, 2415, 2425, 2435, 2445, 7075, 7085, 7095, 7105, 7115, 7125, 7135, 7145, 7155, 7165, 7175, 7185, 7195, 7205, 7215, 7225, 7235, 7245

Offset

1,1

Comments

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

From Marius A. Burtea, Oct 25 2021: (Start)

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, ...

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)

Links

Table of n, a(n) for n=1..44.

Example

75^2 = 5625, hence 75 is a term.
235^2 = 55225, hence 235 is not a term.

Mathematica

Select[5 * Range[2, 1500], (d = IntegerDigits[#^2])[[1]] == d[[-1]] == 5 && d[[2]] != 5 &] (* Amiram Eldar, Oct 25 2021 *)

Prog

(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
(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
(Python)
from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("5")) == len(s.lstrip("5")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+5 for i in count(0)))
  return [k for k in r if ok(k)]
print(aupto(7245)) # Michael S. Branicky, Oct 26 2021

Crossrefs

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

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

Subsequence of A305719.

Sequence in context: A003503 A201916 A098230 * A258056 A174685 A158742

Adjacent sequences: A348486 A348487 A348488 * A348490 A348491 A348492

Keyword

nonn,base

Author

Bernard Schott, Oct 25 2021

Status

approved