A348831 Positive numbers whose square starts and ends with exactly 44, and no 444.

212, 2112, 6638, 6662, 6688, 20988, 21012, 21062, 21112, 21138, 21162, 21188, 21212, 66338, 66362, 66388, 66412, 66438, 66488, 66512, 66562, 66588, 66612, 66712, 66738, 66762, 66788, 66812, 66838, 66862, 66888, 66912, 66938, 66988, 67012, 67062, 209762, 209788

Offset: 1

Author

Bernard Schott, Nov 08 2021

Keywords

• nonn
• base

Comments

When a square starts and ends with digits dd, then dd is necessarily 44.

The last 2 digits of terms are either 12, 38, 62 or 88.

From Marius A. Burtea, Nov 09 2021 : (Start)

The sequence is infinite because the numbers 212, 2112, 21112, ..., (19*10^k + 8) / 9, k >= 3, are terms because the remainder when dividing by 1000 is 544 and 445*10^(2*k - 2) < ((19*10^k + 8) / 9)^2 < 447*10^(2*k - 2), k >= 3.

Also 6638, 66338, 663338, 6633338, 66333338, 663333338, 6633333338, ..., (199*10^k + 14) / 3, k >= 2, are terms and have no digits 0, because their squares are: 44063044, 4400730244, 4400730244, 440017302244, 44001173022244, 4400111730222244, 440011117302222244, ... (End)

Examples

			212 is a term since 212^2 = 44944.
662 is not a term since 662^2 = 438244.
668 is not a term since 668^2 = 446224.
2108 is not a term since 2108^2 = 4443664.
21038 is not a term since 21038^2 = 442597444.
21088 is not a term since 21088^2 = 444703744.
		

Crossrefs

Cf. A017317.

Subsequence of A045858, A273375, A305719 and A346774.

Similar to: A348488 (d=4), this sequence (dd=44), A348832 (ddd=444).

Programs

Magma

fd:=func; fs:=func; [n:n in [1..210000]|fd(n) and fs(n)]; // Marius A. Burtea, Nov 08 2021

Mathematica

Select[Range[10, 300000], (d = IntegerDigits[#^2])[[1 ;; 2]] ==  d[[-2 ;; -1]] == {4, 4} && d[[-3]] != 4 && d[[3]] != 4 &] (* Amiram Eldar, Nov 08 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)-2
def aupto(N):
  ends = [12, 38, 62, 88]
  r = takewhile(lambda x: x<=N, (100*i+d for i in count(0) for d in ends))
  return [k for k in r if ok(k)]
print(aupto(209788)) # Michael S. Branicky, Nov 08 2021