A273375 -id:A273375 - cp's OEIS frontend

A273373 Squares ending in digit 6.

16, 36, 196, 256, 576, 676, 1156, 1296, 1936, 2116, 2916, 3136, 4096, 4356, 5476, 5776, 7056, 7396, 8836, 9216, 10816, 11236, 12996, 13456, 15376, 15876, 17956, 18496, 20736, 21316, 23716, 24336, 26896, 27556, 30276, 30976, 33856, 34596, 37636, 38416, 41616

Offset: 1

Vincenzo Librandi, May 21 2016

These are the only squares whose second last digit is odd. This implies that the only squares whose last two digits are the same are those ending with 0 or 4; those ending with 1, 5, and 9 are paired with even second last digits. - Waldemar Puszkarz, May 24 2016

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A047221, A090773.
Cf. A017341 (numbers ending in 6), A017343 (cubes ending in 6).
Cf. squares with last digit k: A017270 (k=0), A273372 (k=1), A273375 (k=4), A017330 (k=5), this sequence (k=6), A273374 (k=9).

/* By definition: */ [n^2: n in [0..200] | Modexp(n,2,10) eq 6];

[(10*n - 3*(-1)^n - 5)^2/4: n in [1..50]];

seq(seq((10*i+j)^2,j=[4,6]),i=0..20); # Robert Israel, May 24 2016

Table[(10 n - 3 (-1)^n - 5)^2/4, {n, 1, 50}]
CoefficientList[Series[4 (4 + 5 x + 32 x^2 + 5 x^3 + 4 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x]
Select[Range[250]^2,Mod[#,10]==6&] (* Harvey P. Dale, May 31 2020 *)

G.f.: 4*x*(4 + 5*x + 32*x^2 + 5*x^3 + 4*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = 4*A047221(n)^2 = (10*n - 3*(-1)^n - 5)^2/4.
a(n) = A090773(n)^2. - Michel Marcus, May 25 2016
Sum_{n>=1} 1/a(n) = 2*Pi^2/(25*(5+sqrt(5))). - Amiram Eldar, Feb 16 2023

Corrected and extended by Bruno Berselli, May 23 2016

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

Original entry on oeis.org

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

Bernard Schott, Oct 24 2021

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

			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.

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).

[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

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

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

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

A348832 Positive numbers whose square starts and ends with exactly 444.

Original entry on oeis.org

666462, 666538, 666962, 667038, 2107462, 2107538, 2107962, 2108038, 2108462, 2108538, 2108962, 2109038, 2109462, 6663462, 6663538, 6663962, 6664038, 6664462, 6664538, 6664962, 6665038, 6665462, 6665538, 6665962, 6666038, 6667462, 6667538, 6667962, 6668038, 6668462, 6668538, 6668962

Offset: 1

Bernard Schott, Nov 09 2021

The 1st problem of British Mathematical Olympiad (BMO) in 1995 (see link) asked to find all positive integers whose squares end in three 4’s (A039685); this sequence is the subsequence of these integers whose squares also start in precisely three 4's (no four or more 4's). Two such infinite subsequences are proposed below.
When a square starts and ends with digits ddd, then ddd is necessarily 444.
The first 3 digits of terms are either 210, 666 or 667, while the last 3 digits are either 038, 462, 538 or 962 (see examples).
From Marius A. Burtea, Nov 09 2021 : (Start)
The sequence is infinite because the numbers 667038, 6670038, 66700038, 667000038, ..., 667*10^k + 38, k >= 3, are terms because are square 444939693444, 44489406921444, 4448895069201444, 444889050692001444, 44488900506920001444, ...
Also, 6663462, 66633462, 666333462, 6663333462, ..., (1999*10^k + 386) / 3, k >= 4, are terms and have no digits 0, because their squares are 44401725825444, 4440018258105444, 444000282580905444, 44400012825808905444,
4440001128258088905444, ... (End)

			666462 is a term since 666462^2 = 444171597444.
21038 is not a term since 21038^2 = 442597444.

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 55 and 95-96 (1995)

British Mathematical Olympiad 1975, Problem 1.

Cf. A017317, A328886.
Subsequence of A039685, A045858, A273375, A305719, A346892.
Similar to: A348488 (d=4), A348831 (dd=44), this sequence (ddd=444).

fd:=func; fs:=func; [n:n in [1..6700000]|fd(n) and fs(n)]; // Marius A. Burtea, Nov 09 2021

Select[Range[100, 7*10^6], (d = IntegerDigits[#^2])[[1 ;; 3]] == d[[-3 ;; -1]] == {4, 4, 4} && d[[-4]] != 4 && d[[4]] != 4 &] (* Amiram Eldar, Nov 09 2021 *)

from itertools import count, takewhile
def ok(n):
  s = str(n*n); return len(s.rstrip("4")) == len(s.lstrip("4")) == len(s)-3
def aupto(N):
  ends = [38, 462, 538, 962]
  r = takewhile(lambda x: x<=N, (1000*i+d for i in count(0) for d in ends))
  return [k for k in r if ok(k)]
print(aupto(6668962)) # Michael S. Branicky, Nov 09 2021

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

Original entry on oeis.org

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

Bernard Schott, Nov 08 2021

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)

			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.

Cf. A017317.
Subsequence of A045858, A273375, A305719 and A346774.
Similar to: A348488 (d=4), this sequence (dd=44), A348832 (ddd=444).

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

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 *)

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

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A273373 Squares ending in digit 6.

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

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A348832 Positive numbers whose square starts and ends with exactly 444.

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

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