A339676 -id:A339676 - cp's OEIS frontend

A340549 Smallest integer with exactly n divisors that are repunits.

1, 11, 1111, 111111, 11222211, 111111111111, 1111222222221111, 11223344555544332211, 112244668899998866442211, 112357025813567765307519653211, 112244781144780011109977441077442211, 113491945266228931047738906599340328084311, 113378566812907968345622215431647587096554773311

Offset: 1

Bernard Schott, Jan 12 2021

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.
From Bernard Schott, Jan 13 2022: (Start)
Repunit terms are: R_1, R_2, R_4, R_6, R_12, ... where R_m is A002275(m).
It appears that palindromes occur for n = 1 to 9 only. (End)
The indices of the n repunits that divide a(n) are given by the n-th row of A356184. - Bernard Schott, Sep 13 2022

			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.
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.

Michel Marcus, Table of n, a(n) for n = 1..18

Cf. A002275, A083230, A083278, A308365, A339676, A356184.

Similar, but with divisors that are: A087997 (palindromes), A355699 (repdigits).

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 *)

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 }
process(n, i) = { if(n <=ulim, listput(l, n); for(j = i + 1, #reps, c = lcm(n, reps[j]); process(c, j) ) ) }
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

a(5)-a(13) from David A. Corneth, Jan 12 2021

Definition modified by Bernard Schott, Sep 05 2022

A308365 Numbers which are products of repunits.

Original entry on oeis.org

1, 11, 111, 121, 1111, 1221, 1331, 11111, 12221, 12321, 13431, 14641, 111111, 122221, 123321, 134431, 135531, 147741, 161051, 1111111, 1222221, 1233321, 1234321, 1344431, 1356531, 1367631, 1478741, 1490841, 1625151, 1771561, 11111111, 12222221, 12333321

Offset: 1

Sergio Pimentel, May 22 2019

The number of terms below 10^n is A216053(n)-1 for 1 <= n <= 25, but not for larger n. - Rémy Sigrist, May 28 2019

The product of repunits is not necessarily palindromic, see A339676. - Bernard Schott, Apr 02 2021

			a(11) = 13431 is in the sequence since it is the product of repunits (11^2*111).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002275 (repunits), A083278 (repunit powers), A216053, A339676 (nonpalindromic terms).

d:= 10: # for terms < 10^d
N:= 10^d:
S:= {1}:
for m from 2 to d do
  r:= (10^m-1)/9;
  k:= floor(log[r](N));
  V:= S;
  for i from 1 to k do
    V:= select(`<`,map(`*`,V,r),N);
    S:= S union V
  od;
od:
sort(convert(S,list)); # Robert Israel, Nov 26 2020

Missing a(25) = 1356531 inserted by Ilya Gutkovskiy, Apr 14 2020

cp's OEIS Frontend

A340549 Smallest integer with exactly n divisors that are repunits.

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