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 A348832 Positive numbers whose square starts and ends with exactly 444. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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) REFERENCES 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) LINKS Table of n, a(n) for n=1..32. British Mathematical Olympiad 1975, Problem 1. EXAMPLE 666462 is a term since 666462^2 = 444171597444. 21038 is not a term since 21038^2 = 442597444. MATHEMATICA 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 *) PROG (Python) 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 (Magma) fd:=func; fs:=func; [n:n in [1..6700000]|fd(n) and fs(n)]; // Marius A. Burtea, Nov 09 2021 CROSSREFS Cf. A017317, A328886. Subsequence of A039685, A045858, A273375, A305719, A346892. Similar to: A348488 (d=4), A348831 (dd=44), this sequence (ddd=444). Sequence in context: A144132 A069373 A270801 * A162840 A233817 A204887 Adjacent sequences: A348829 A348830 A348831 * A348833 A348834 A348835 KEYWORD nonn,base AUTHOR Bernard Schott, Nov 09 2021 STATUS approved

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Last modified June 17 13:47 EDT 2024. Contains 373445 sequences. (Running on oeis4.)