cp's OEIS Frontend

A309062 Oblong numbers that are repdigits with length > 2 in more than two bases.

%I A309062 #88 Nov 23 2019 04:07:24
%S A309062 61035156,641431602,38146972656,70607384120,953674316406,
%T A309062 5824521280620,23841857910156,51472783023662,145655559307440,
%U A309062 463255047212960,1838956877846660,14901161193847656,37523658824249780,88453695801367260,166354152295794960,416972378738246240
%N A309062 Oblong numbers that are repdigits with length > 2 in more than two bases.
%C A309062 All initial terms come from the b-file in A290869.
%C A309062 For the given terms, the number of bases are respectively 4, 3, 3, 4, 4, 4, 4, 3, 4, 3 and 4.
%C A309062 A003463(64), A003463(24) (confirmed) and A003463(36) are candidates for 5, 6 and 7 bases representations.
%C A309062 From _Bernard Schott_, Jul 24 2019: (Start)
%C A309062 The terms of this sequence are necessarily of the form (b^(2*q) - 1)/4 with q > 2 and b = 4*m+1 with m > 0, but when b = c^2 is an odd square (A016754), then some terms can also have the form (b^(2*q+1) - 1)/4 as a(8) and a(23). If these terms have representations in u bases, the values of (b, 2*q or 2*q+1, u) for the first eleven terms are respectively (5, 12, 4), (37, 6, 3), (5, 16, 3), (9, 12, 4), (5, 18, 4), (13, 12, 4), (5, 20, 4), (9, 15, 3), (17, 12, 4), (9, 16, 3) and (21, 12, 4).
%C A309062 For any b = 4*m+1 with m > 0 and r > 2, (b^(4*r) - 1)/4 is an oblong repdigit with length > 2 in at least bases b, b^2 and b^4; hence this sequence is infinite.
%C A309062 (End)
%C A309062 From _Chai Wah Wu_, Jul 24 2019: (Start)
%C A309062 Other values of (b, q, u) for which (b^(2*q) - 1)/4 is a term with representations in u bases:
%C A309062 (5, 12, 6), (5, 14, 4), (5, 15, 6), (9, 9, 4), (9, 10, 4), (13, 8, 3), (13, 9, 4), (17, 8, 3), (29, 6, 4), (33, 6, 4), (37, 6, 4), (41, 6, 4), (45, 6, 4).
%C A309062 (End)
%C A309062 From _Bernard Schott_, Jul 24 2019: (Start)
%C A309062 Theorem: if tau(2*q) = r > 4, (b^(2*q) - 1)/4 is a term that has exactly r-2 representations as repdigits with length > 2 in bases that are powers of b.
%C A309062 There exist cases where a term also has representation in another base that is not power of b. For instance a(2), see example, where base 3446 is not a perfect power of 37.
%C A309062 Conclusion: if m = (b^(2*q) - 1)/4 is a term and if beta"(m) is the number of representations of this term as repdigits with length > 2, then, beta"(m) >= tau(2*q) - 2. (End)
%H A309062 Giovanni Resta, <a href="/A309062/b309062.txt">Table of n, a(n) for n = 1..26</a>
%e A309062 From _Bernard Schott_, Jul 18 2019: (Start)
%e A309062 a(1) = 61035156 = 7812*7813 = 111111111111_5 = 666666_25 = (31,31,31)_125 = (156,156,156)_625.
%e A309062 a(2) = 641431602 = 25326*25327 = 999999_37 = (342,342,342)_1469 = (54,54,54)_3446.
%e A309062 (End)
%e A309062 a(11) = 1838956877846660 = 42883060*42883061 = 555555555555_21 = (110, 110, 110, 110, 110, 110)_441 = (2315, 2315, 2315, 2315)_9261 = (48620, 48620, 48620)_194481. - _Chai Wah Wu_, Jul 24 2019
%o A309062 (PARI) isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
%o A309062 okrepu3(b, target, lim) = {my(k = 3, nb = 0, x); while ((x=(b^k-1)/(b-1)) <= target, if (x==target, nb++); k++); nb;}
%o A309062 dge3(n) = {my(d=divisors(n), nb=0, ndi, limi); for (i=1, #d, ndi = n/d[i]; limi = sqrtint(ndi); for (k=d[i]+1, limi, nb += okrepu3(k, ndi, limi););); nb;}
%o A309062 isok(n) = isoblong(n) && (dge3(n) >= 3);
%Y A309062 Intersection of A002378 and A290869.
%Y A309062 Cf. A326378 (similar in no base), A326384 (similar in one base), A326385 (similar in 2 bases).
%K A309062 nonn,base
%O A309062 1,1
%A A309062 _Michel Marcus_, Jul 10 2019
%E A309062 a(11) from _Chai Wah Wu_, Jul 21 2019
%E A309062 a(12)-a(16) from _Giovanni Resta_, Jul 28 2019