A083230 -id:A083230 - 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

A356018 a(n) is the number of evil divisors (A001969) of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 0, 2, 2, 0, 3, 0, 0, 3, 0, 1, 4, 0, 3, 1, 0, 1, 4, 1, 0, 3, 0, 1, 6, 0, 0, 2, 2, 1, 6, 0, 0, 2, 4, 0, 2, 1, 0, 5, 2, 0, 5, 0, 2, 3, 0, 1, 6, 1, 0, 2, 2, 0, 9, 0, 0, 3, 0, 2, 4, 0, 3, 2, 2, 1, 8, 0, 0, 4, 0, 1, 4, 0, 5, 3, 0, 1, 3, 3, 2

Offset: 1

Bernard Schott, Jul 23 2022

a(n) = 0 iff n is in A093696.

			12 has 6 divisors: {1, 2, 3, 4, 6, 12} of which three {3, 6, 12} have an even number of 1's in their binary expansion with 11, 110 and 11100 respectively; hence a(12) = 3.

Cf. A000005, A001969, A093688, A093696 (location of 0s), A227872, A356019, A356020.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

A356018 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isA001969(d) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A356018(n),n=1..200) ;  # R. J. Mathar, Aug 07 2022

a[n_] := DivisorSum[n, 1 &, EvenQ[DigitCount[#, 2, 1]] &]; Array[a, 100] (* Amiram Eldar, Jul 23 2022 *)

a(n) = my(v = valuation(n, 2)); n>>=v; d=divisors(n); sum(i=1, #d, bitand(hammingweight(d[i]), 1) == 0) * (v+1) \\ David A. Corneth, Jul 23 2022

from sympy import divisors
def c(n): return bin(n).count("1")&1 == 0
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 23 2022

a(n) = A000005(n) - A227872(n).

More terms from David A. Corneth, Jul 23 2022

A043284 Maximal run length in base-10 representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2

Offset: 1

Clark Kimberling

The first term larger than 2 is a(111) = 3. - M. F. Hasler, Jul 21 2013

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A043276-A043290 for base-2 to base-16 analogs.
Cf. A030556-A030561, A030575-A030580 (related to base-6 run lengths).
Cf. A227186, A227188, A101211, A005811 (related to base-2 run lengths).

A043284[n_]:=Max[Map[Length,Split[IntegerDigits[n]]]];Array[A043284,100] (* Paolo Xausa, Sep 27 2023 *)

A043284(n)={my(m,c=1);while(n>0,n%10==(n\=10)%10 && c++ && next;m=max(m,c);c=1);m} \\ M. F. Hasler, Jul 23 2013

For n < 111, a(n) = 1 except for a(n) = 2 when n==0 (mod 11) or n = 100. - M. F. Hasler, Jul 21 2013

Data completed up to a(100), first difference with A083230, by M. F. Hasler, Oct 18 2019

A355770 a(n) is the number of terms of A333369 that divide n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 2, 2, 2, 4, 1, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 4, 2, 1, 2, 2, 4, 3, 2, 2, 4, 2, 1, 3, 1, 3, 5, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, 2, 4, 1, 2, 4, 1, 2, 4, 1, 3, 4, 1, 2, 2, 4, 2, 3, 2, 2, 5, 2, 2, 4, 2, 2, 3, 1, 1, 3, 3, 1, 2

Offset: 1

Bernard Schott, Jul 16 2022

Cf. A333369, A353735, A355771, A355772, A355773.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

q[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 16 2022 *)

issimber(m) = my(d=digits(m), s=Set(d)); for (i=1, #s, if (#select(x->(x==s[i]), d) % 2 != (s[i] % 2), return (0))); return (1); \\ A333369
a(n) = sumdiv(n, d, issimber(d)); \\ Michel Marcus, Jul 18 2022

from sympy import divisors
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jul 16 2022

More terms from Michael S. Branicky, Jul 16 2022

A355698 a(n) is the number of repdigits divisors of n (A010785).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 2, 5, 1, 3, 3, 4, 1, 5, 1, 4, 3, 4, 1, 6, 2, 2, 3, 4, 1, 5, 1, 4, 4, 2, 3, 6, 1, 2, 2, 5, 1, 5, 1, 6, 4, 2, 1, 6, 2, 3, 2, 3, 1, 5, 4, 5, 2, 2, 1, 6, 1, 2, 4, 4, 2, 8, 1, 3, 2, 4, 1, 7, 1, 2, 3, 3, 4, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 8, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 6, 4, 1, 4, 1, 4, 4

Offset: 1

Bernard Schott, Jul 14 2022

More than the usual number of terms are displayed in order to show the difference from A087990.
The first 100 terms are the same first 100 terms of A087990, then a(101) = 1 while A087990(101) = 2, because 101 is the smallest palindrome that is not repdigit; the next difference is 121.
Inequalities: 1 <= a(n) <= A087990(n).

			66 has 8 divisors: {1, 2, 3, 6, 11, 22, 33, 66} that are all repdigits, hence a(66) = 8.
121 has 3 divisors: {1, 11, 121} of which 2 are repdigits: {1, 11}, hence a(121) = 2.

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

Cf. A010785, A065444, A087990, A190217, A340548, A355699.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

isrepdig:= proc(n) nops(convert(convert(n,base,10),set))=1 end proc:
f:= proc(n) nops(select(isrepdig, numtheory:-divisors(n))) end proc:
map(f, [$1..200]); # Robert Israel, Aug 07 2024

a[n_] := DivisorSum[n, 1 &, Length[Union[IntegerDigits[#]]] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 14 2022 *)

a(n) = my(ret=0,u=1); while(u<=n, ret+=sum(d=1,9, n%(u*d)==0); u=10*u+1); ret; \\ Kevin Ryde, Jul 14 2022

isrep(n) = {1==#Set(digits(n))}; \\ A010785
a(n) = sumdiv(n, d, isrep(d)); \\ Michel Marcus, Jul 15 2022

from sympy import divisors
def c(n): return len(set(str(n))) == 1
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Jul 14 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (7129/2520) * A065444 = 3.11446261209177581335... . - Amiram Eldar, Apr 17 2025

A356184 Triangle read by rows: n-th row gives the indices of the n repunits that divide A340549(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 6, 1, 2, 3, 4, 6, 1, 2, 3, 4, 6, 12, 1, 2, 3, 4, 6, 8, 12, 1, 2, 3, 4, 5, 6, 10, 12, 1, 2, 3, 4, 5, 6, 8, 10, 12, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24

Offset: 1

Bernard Schott, Jul 28 2022

			Triangle begins:
  1;
  1,  2;
  1,  2,  4;
  1,  2,  3,  6;
  1,  2,  3,  4,  6;
  1,  2,  3,  4,  6, 12;
  1,  2,  3,  4,  6,  8, 12;
  1,  2,  3,  4,  5,  6, 10, 12;
  1,  2,  3,  4,  5,  6,  8, 10, 12;
  1,  2,  3,  4,  5,  6,  8,  9, 10, 12;
  1,  2,  3,  4,  5,  6,  8,  9, 10, 12, 18;
  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 12, 14;
  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 12, 14, 18;
  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 12, 14, 15, 18;
  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 12, 14, 15, 18, 20;
  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 12, 14, 15, 18, 20, 24;
  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 12, 14, 15, 16, 18, 20, 24;
  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 12, 14, 15, 16, 18, 20, 24, 30;
  ...
The 5th row is {1, 2, 3, 4, 6} since A340549(5) = 11222211 is the least integer that is divisible by five repunits and these are R_1, R_2, R_3, R_4 and R_6.

Cf. A002275, A004023, A083230, A340549.

cp's OEIS Frontend

A340549 Smallest integer with exactly n divisors that are repunits.

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A356018 a(n) is the number of evil divisors (A001969) of n.

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A043284 Maximal run length in base-10 representation of n.

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A355770 a(n) is the number of terms of A333369 that divide n.

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A355698 a(n) is the number of repdigits divisors of n (A010785).

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A356184 Triangle read by rows: n-th row gives the indices of the n repunits that divide A340549(n).

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