A308365 -id:A308365 - 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

A334131 Numbers that can be written as a product of distinct repunits.

Original entry on oeis.org

0, 1, 11, 111, 1111, 1221, 11111, 12221, 111111, 122221, 123321, 1111111, 1222221, 1233321, 1356531, 11111111, 12222221, 12333321, 12344321, 13566531, 111111111, 122222221, 123333321, 123444321, 135666531, 135787531, 1111111111, 1222222221

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

			13566531 = 11*111*11111. - _David A. Corneth_, Mar 26 2021

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A002275, A015009, A077488, A083278, A308365.

A339676 Nonpalindromic numbers that are products of repunits.

Original entry on oeis.org

161051, 1490841, 1625151, 1771561, 14921841, 15043941, 16266151, 16399251, 17876661, 19487171, 137009631, 149231841, 149352841, 150574941, 151807041, 162676151, 164140251, 165483351, 178927661, 180391761, 196643271, 214358881, 1370219631, 1371330631, 1492331841

Offset: 1

Bernard Schott, Dec 12 2020

The first term is A308365(19).

G. J. Simmons conjectured there are no palindromes of form n^k for k >= 5 (and n > 1) (see link, page 98). According to this conjecture, these perfect powers are terms: {11^k, k>=4}, {111^k, k>=4}, {1111^k, k>=3}, {11111^k, k>=3}, ...

			a(1) = 161051 = 11^5.
a(2) = 1490841 = 11^2 * 111^2.
a(3) = 1625151 = 11^4 * 111.
a(4) = 1771561 = 11^6.
a(5) = 14921841 = 11^2 * 111 * 1111.

David A. Corneth, Table of n, a(n) for n = 1..10000
Gustavus J. Simmons, Palindromic Powers, J. Rec. Math., 3 (No. 2, 1970), 93-98 [Annotated scanned copy].

Intersection of A308365 and A029742.

Cf. A083278, A334131.

vec[max_] := Module[{m = Floor @ Log10[9*max + 1], r, s = {1}, s1}, r = (10^Range[2, m] - 1)/9; Do[emax = Floor@Log[r[[k]], max]; s1 = r[[k]]^Range[0, emax]; s = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &], {k, 1, m - 1}]; s]; Select[vec[1.5*10^9], !PalindromeQ[#] &] (* Amiram Eldar, Dec 12 2020 *)

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A340549 Smallest integer with exactly n divisors that are repunits.

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A334131 Numbers that can be written as a product of distinct repunits.

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A339676 Nonpalindromic numbers that are products of repunits.

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