A348488 Positive numbers whose square starts and ends with exactly one 4.

2, 22, 68, 202, 208, 218, 222, 642, 648, 652, 658, 672, 678, 682, 692, 698, 702, 2002, 2008, 2018, 2022, 2028, 2032, 2042, 2048, 2052, 2058, 2068, 2072, 2078, 2082, 2092, 2122, 2128, 2132, 2142, 2148, 2152, 2158, 2168, 2172, 2178, 2182, 2192, 2198, 2202, 2208, 2218, 2222, 2228

Offset

1,1

Comments

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

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

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

Links

Table of n, a(n) for n=1..50.

Example

22 is a term since 22^2 = 484.
638 is not a term since 638^2 = 407044.
668 is not a term since 668^2 = 446224.

Mathematica

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

Prog

(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
(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
(Python)
from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-1
def aupto(N):
  r = takewhile(lambda x: x<=N, (10*i+d for i in count(0) for d in [2, 8]))
  return [k for k in r if ok(k)]
print(aupto(2228)) # Michael S. Branicky, Oct 24 2021

Crossrefs

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

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

Subsequence of A305719.

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

Sequence in context: A273366 A053940 A156480 * A226406 A226335 A123061

Adjacent sequences: A348485 A348486 A348487 * A348489 A348490 A348491

Keyword

nonn,base

Author

Bernard Schott, Oct 24 2021

Status

approved