

A309062


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


7



61035156, 641431602, 38146972656, 70607384120, 953674316406, 5824521280620, 23841857910156, 51472783023662, 145655559307440, 463255047212960, 1838956877846660, 14901161193847656, 37523658824249780, 88453695801367260, 166354152295794960, 416972378738246240
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OFFSET

1,1


COMMENTS

All initial terms come from the bfile in A290869.
For the given terms, the number of bases are respectively 4, 3, 3, 4, 4, 4, 4, 3, 4, 3 and 4.
A003463(64), A003463(24) (confirmed) and A003463(36) are candidates for 5, 6 and 7 bases representations.
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).
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.
(End)
Other values of (b, q, u) for which (b^(2*q)  1)/4 is a term with representations in u bases:
(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).
(End)
Theorem: if tau(2*q) = r > 4, (b^(2*q)  1)/4 is a term that has exactly r2 representations as repdigits with length > 2 in bases that are powers of b.
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.
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)


LINKS



EXAMPLE

a(1) = 61035156 = 7812*7813 = 111111111111_5 = 666666_25 = (31,31,31)_125 = (156,156,156)_625.
a(2) = 641431602 = 25326*25327 = 999999_37 = (342,342,342)_1469 = (54,54,54)_3446.
(End)
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


PROG

(PARI) isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
okrepu3(b, target, lim) = {my(k = 3, nb = 0, x); while ((x=(b^k1)/(b1)) <= target, if (x==target, nb++); k++); nb; }
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; }
isok(n) = isoblong(n) && (dge3(n) >= 3);


CROSSREFS



KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



