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

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

Offset: 1

Author

Michel Marcus, Jul 10 2019

Keywords

• nonn
• base

Comments

All initial terms come from the b-file 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.

From Bernard Schott, Jul 24 2019: (Start)

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)

From Chai Wah Wu, Jul 24 2019: (Start)

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)

From Bernard Schott, Jul 24 2019: (Start)

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.

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)

Examples

			From _Bernard Schott_, Jul 18 2019: (Start)
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
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..26

Crossrefs

Intersection of A002378 and A290869.

Cf. A326378 (similar in no base), A326384 (similar in one base), A326385 (similar in 2 bases).

Programs

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^k-1)/(b-1)) <= 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);

Extensions

a(11) from Chai Wah Wu, Jul 21 2019

a(12)-a(16) from Giovanni Resta, Jul 28 2019