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A340549 Smallest integer with exactly n divisors that are repunits.

%I A340549 #91 Sep 16 2022 05:32:47
%S A340549 1,11,1111,111111,11222211,111111111111,1111222222221111,
%T A340549 11223344555544332211,112244668899998866442211,
%U A340549 112357025813567765307519653211,112244781144780011109977441077442211,113491945266228931047738906599340328084311,113378566812907968345622215431647587096554773311
%N A340549 Smallest integer with exactly n divisors that are repunits.
%C A340549 Previous name was: Integers whose number of divisors that are repunits sets a new record. From a(1) up to a(18), the terms of these two sequences are exactly the same.
%C A340549 From _Bernard Schott_, Jan 13 2022: (Start)
%C A340549 Repunit terms are: R_1, R_2, R_4, R_6, R_12, ... where R_m is A002275(m).
%C A340549 It appears that palindromes occur for n = 1 to 9 only. (End)
%C A340549 The indices of the n repunits that divide a(n) are given by the n-th row of A356184. - _Bernard Schott_, Sep 13 2022
%H A340549 Michel Marcus, <a href="/A340549/b340549.txt">Table of n, a(n) for n = 1..18</a>
%e A340549 111111 has 4 divisors that are repunits: {1, 11, 111, 111111}; also, 111111 is the smallest integer that has at least 4 repunit divisors, hence 111111 is a term.
%e A340549 The 13 repunit divisors of a(13) are R_1, R_2, R_3, R_4, R_5, R_6, R_7, R_8, R_9, R_10, R_12, R_14 and R_18.
%t A340549 repQ[n_] := Union @ IntegerDigits[n] == {1}; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = DivisorSum[n, 1 &, repQ[#] &]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[4, 10^7] (* _Amiram Eldar_, Sep 05 2022 *)
%o A340549 (PARI) upto(n) = { l = List(); ulim = n; res = []; reps = vector(logint(n, 10)-1, i, 10^(i+1)\9); for(i = 0, #reps, process(1, i); ); listsort(l, 1); r = 0; for(i = 1, #l, c = f(l[i]); if(c > #res, res = concat(res, vector(c - #res, j, oo)); ); res[c] = min(res[c], l[i]) ); res }
%o A340549 process(n, i) = { if(n <=ulim, listput(l, n); for(j = i + 1, #reps, c = lcm(n, reps[j]); process(c, j) ) ) }
%o A340549 f(n) = my(u = logint(n, 10) + 2); 1 + sum(i = 1, u, n % (10^(i+1)\9) == 0) \\ _David A. Corneth_, Jan 12 2021, Jan 17 2022, Sep 12 2022
%Y A340549 Cf. A002275, A083230, A083278, A308365, A339676, A356184.
%Y A340549 Similar, but with divisors that are: A087997 (palindromes), A355699 (repdigits).
%K A340549 nonn,base
%O A340549 1,2
%A A340549 _Bernard Schott_, Jan 12 2021
%E A340549 a(5)-a(13) from _David A. Corneth_, Jan 12 2021
%E A340549 Definition modified by _Bernard Schott_, Sep 05 2022