A340548 Integers whose number of repdigit divisors sets a new record.

1, 2, 4, 6, 12, 24, 66, 132, 264, 792, 3960, 14652, 26664, 29304, 79992, 146520, 399960, 1025640, 2666664, 7999992, 13333320, 39999960, 269333064, 807999192, 1346665320, 4039995960, 28279971720, 7999999999992, 8080799919192, 13333333333320, 13467999865320, 39999999999960, 40403999595960

Offset: 1

Author

Bernard Schott, Jan 11 2021

Keywords

• nonn
• base

Comments

The first 10 terms are the same as A093036, then A093036(11) = 1848 while a(11) = 3960, because from a(1) to a(10), all palindromic divisors are also repdigits, and then 616 is a non-repdigit palindromic divisor of 1848.

Number of repdigit divisors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 17, 18, ...

Indices of repdigits: 1, 2, 3, 4, 7, ...

Examples

			132 has 12 divisors: {1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132} of which 10 are repdigits: {1, 2, 3, 4, 6, 11, 22, 33, 44, 66}. No positive integer smaller than 132 has as many as ten repdigit divisors; hence 132 is a term.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..53 (terms <= 10^30).

Crossrefs

Cf. A010785, A190217.

Similar for: A053624 (odd), A181808 (even), A093036 (palindromes), A340549 (repunits).

Programs

Mathematica

repQ[n_] := Length @ Union @ IntegerDigits[n] == 1; s[n_] := DivisorSum[n, 1 &, repQ[#] &]; smax =  0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jan 11 2021 *)

PARI

isrd(n) = {1 == #Set(digits(n))}; \\ A010785
f(n) = sumdiv(n, d, isrd(d));
lista(nn) = {my(m = 0); for (n=1, nn, my(x = f(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 11 2021

Extensions

a(16)-a(20) from Michel Marcus, Jan 11 2021

a(21)-a(26) from Amiram Eldar, Jan 12 2021

a(27) from Chai Wah Wu, Jan 14 2021

More terms from David A. Corneth, Jan 15 2021