A087997 -id:A087997 - 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

A355699 a(n) is the smallest number that has exactly n repdigit divisors.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 72, 66, 666, 132, 1332, 264, 2664, 792, 13320, 3960, 14652, 26664, 48840, 29304, 79992, 341880, 146520, 399960, 1333332, 1025640, 2799720, 8879112, 2666664, 18666648, 7999992, 44395560, 13333320, 93333240, 39999960, 279999720, 269333064

Offset: 1

Bernard Schott, Jul 14 2022

			72 has 12 divisors: {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, only {1, 2, 3, 4, 6, 8, 9} are repdigits; no positive integer smaller than 72 has seven repdigit divisors, hence a(7) = 72.

David A. Corneth, Table of n, a(n) for n = 1..135
David A. Corneth, PARI program

Cf. A010785, A087990, A190217, A340548, A355698.

Similar sequences: A087997, A333456, A355303, A355695.

f[n_] := DivisorSum[n, 1 &, Length[Union[IntegerDigits[#]]] == 1 &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[24, 10^6] (* Amiram Eldar, Jul 15 2022 *)

isrep(n) = 1==#Set(digits(n)); \\ A010785
a(n) = my(k=1); while (sumdiv(k, d, isrep(d)) != n, k++); k; \\ Michel Marcus, Jul 15 2022

\\ See PARI link. - David A. Corneth, Jul 26 2022

from sympy import divisors
from itertools import count, islice
def c(n): return len(set(str(n))) == 1
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 21))) # Michael S. Branicky, Jul 26 2022

a(9)-a(35) from Michael S. Branicky, Jul 14 2022

a(36)-a(37) from Michael S. Branicky, Jul 15 2022

A355771 a(n) is the smallest integer that has exactly n divisors from A333369.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 195, 315, 945, 900, 1575, 2100, 3900, 6825, 11655, 10500, 6300, 18900, 25200, 35100, 27300, 31500, 44100, 94500, 157500, 107100, 81900, 233100, 220500, 598500, 245700, 333900, 409500, 491400, 900900, 573300, 600600, 1228500, 1669500, 1965600

Offset: 1

Bernard Schott, Jul 17 2022

			15 has 4 divisors: {1, 3, 5, 15} all of which are in A333369 integers, and no smaller number has this property, hence a(4) = 15.

Michael S. Branicky, Table of n, a(n) for n = 1..87

Cf. A333369, A353735, A355770, A355772, A355773.
Cf. A005231, A053624.
Similar sequences: A087997, A333456, A355303, A355699.

q[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; f[n_] := DivisorSum[n, 1 &, q[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[40, 10^7] (* Amiram Eldar, Jul 17 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) = my(k=1); while (sumdiv(k, d, issimber(d)) != n, k++); k; \\ Michel Marcus, Jul 18 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 29))) # Michael S. Branicky, Jul 23 2022

More terms from Amiram Eldar, Jul 17 2022

A355968 a(n) is the smallest number that has exactly n odious divisors (A000069).

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 64, 56, 84, 112, 1024, 168, 4096, 448, 336, 728, 36309, 672, 57057, 1456, 1344, 7168, 105105, 2184, 6384, 24150, 5376, 5208, 405405, 4368, 389025, 11648, 20020, 72618, 10416, 8736, 927675, 114114, 48300, 24024, 855855, 17472, 1426425, 40040

Offset: 1

Bernard Schott, Jul 21 2022

a(n) <= 2^(n-1) with equality for n = 1, 2, 3, 4, 5, 7, 11, 13 up to a(44).

			a(6) = 28 since 28 has 6 divisors {1, 2, 4, 7, 14, 28} that have all an odd number of 1's in their binary expansion: 1, 10, 100, 111, 1110 and 11100; also, no positive integer smaller than 28 has six divisors that are odious.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000069, A093696, A227872, A330289, A355969.

Similar sequences: A087997, A333456, A355303, A355699.

f[n_] := DivisorSum[n, 1 &, OddQ[DigitCount[#, 2, 1]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[20, 10^6] (* Amiram Eldar, Jul 21 2022 *)

isod(n) = hammingweight(n) % 2; \\ A000069
a(n) = my(k=1); while (sumdiv(k, d, isod(d)) != n, k++); k; \\ Michel Marcus, Jul 22 2022

from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 36))) # Michael S. Branicky, Jul 25 2022

More terms from Amiram Eldar, Jul 21 2022

A356019 a(n) is the smallest number that has exactly n evil divisors (A001969).

Original entry on oeis.org

1, 3, 6, 12, 18, 45, 30, 135, 72, 60, 90, 765, 120, 1575, 270, 180, 600, 3465, 480, 13545, 360, 540, 1530, 10395, 1260, 720, 3150, 1980, 1080, 49725, 1440, 45045, 2520, 3060, 6930, 2160, 3780, 58905, 27090, 6300, 5040, 184275, 4320, 135135, 6120, 7920, 20790, 329175, 7560, 8640

Offset: 0

Bernard Schott, Jul 23 2022

Differs from A327328 since a(7) = 135 while A327328(7) = 105.

			a(4) = 18 since 18 has six divisors: {1, 2, 3, 6, 9, 18} of which four {3, 6, 9, 18} have an even number of 1's in their binary expansion: 11, 110, 1001 and 10010 respectively; also, no positive integer smaller than 18 has exactly four divisors that are evil.

David A. Corneth, Table of n, a(n) for n = 0..396
David A. Corneth, Upperbounds on a(n), terms <= 8*10^9 are certain

Cf. A001969, A093688, A227872, A327328, A356018, A356020, A356040.

Similar sequences: A087997, A333456, A355303, A355699, A355968.

# output in unsorted b-file style
A356019_list := [seq(0,i=1..1000)] ;
for n from 1 do
    evd := A356018(n) ;
    if evd < nops(A356019_list) then
        if op(evd+1,A356019_list) <= 0 then
            printf("%d %d\n",evd,n) ;
            A356019_list := subsop(evd+1=n,A356019_list) ;
        end if;
    end if;
end do:  # R. J. Mathar, Aug 07 2022

f[n_] := DivisorSum[n, 1 &, EvenQ[DigitCount[#, 2, 1]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^6] (* Amiram Eldar, Jul 23 2022 *)

from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1 == 0
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 0, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Jul 23 2022

a(n) <= A356040(n). - David A. Corneth, Jul 26 2022

More terms from Amiram Eldar, Jul 23 2022

A340796 a(n) is the smallest number with exactly n divisors that are Brazilian.

Original entry on oeis.org

1, 7, 14, 24, 40, 48, 60, 84, 140, 144, 120, 168, 252, 700, 240, 336, 560, 360, 420, 672, 1120, 2304, 960, 720, 1008, 1080, 840, 2184, 1800, 1260, 2016, 5376, 8960, 2160, 1680, 2880, 4032, 3600, 7056, 19600, 3960, 2520, 3360, 6480, 9072, 9900, 6300, 11520, 16128

Offset: 0

Bernard Schott, Jan 21 2021

Primes can be partitioned into Brazilian primes and non-Brazilian primes. If two distinct primes each larger than 11 are in the same category then the larger one has a multiplicity that is smaller than or equal to that of the smaller prime. - David A. Corneth, Jan 24 2021

			Of the eight divisors of 24, three are Brazilian numbers: 8, 12 and 24, and there is no smaller number with three Brazilian divisors, hence a(3) = 24.

David A. Corneth, Table of n, a(n) for n = 0..427
David A. Corneth, More terms
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A085104, A125134, A190300, A220570, A308851, A340795, A340797.

Similar with: A087997 (palindromes), A333456 (Niven), A335038 (Zuckerman).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; d[n_] := DivisorSum[n, 1 &, brazQ[#] &]; m = 30; s = Table[0, {m}]; c = 0; n = 1; While[c < m, i = d[n]; If[i < m && s[[i + 1]] == 0, c++; s[[i + 1]] = n]; n++]; s (* Amiram Eldar, Jan 21 2021 *)

isokb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
isok(k, n) = sumdiv(k, d, isokb(d)) == n;
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jan 23 2021

More terms from Amiram Eldar, Jan 21 2021

A355695 a(n) is the smallest number that has exactly n nonpalindromic divisors (A029742).

Original entry on oeis.org

1, 10, 20, 30, 48, 72, 60, 140, 144, 120, 210, 180, 300, 240, 560, 504, 360, 420, 780, 1764, 900, 960, 720, 1200, 840, 1560, 2640, 1260, 1440, 2400, 3900, 3024, 1680, 3120, 2880, 4800, 7056, 3600, 2520, 3780, 3360, 5460, 6480, 16848, 6300, 8820, 7200, 9240, 6720, 12480, 5040

Offset: 0

Bernard Schott, Jul 14 2022

			48 has 10 divisors: {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}, only 12, 16, 24 and 48 are nonpalindromic; no positive integer smaller than 48 has four nonpalindromic divisors, hence a(4) = 48.

Michael S. Branicky, Table of n, a(n) for n = 0..710

Cf. A029742, A087991, A093037, A334391.

Similar sequences: A087997, A333456, A355303, A355594.

f[n_] := DivisorSum[n, 1 &, ! PalindromeQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^5] (* Amiram Eldar, Jul 14 2022 *)

isnp(n) = my(d=digits(n)); d!=Vecrev(d); \\ A029742
a(n) = my(k=1); while (sumdiv(k, d, isnp(d)) != n, k++); k; \\ Michel Marcus, Jul 14 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return s != s[::-1]
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 0, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 51))) # Michael S. Branicky, Jul 27 2022

More terms from Michel Marcus, Jul 14 2022

A356062 a(n) is the smallest integer that has exactly n Lucas divisors (A000032).

Original entry on oeis.org

1, 2, 4, 12, 36, 252, 2772, 52668, 1211364, 35129556, 1089016236, 44649665676, 2098534286772, 417608323067628, 88115356167269508, 24760415083002731748, 7948093241643876891108, 4140956578896459860267268

Offset: 1

Bernard Schott, Jul 25 2022

The new Lucas numbers that appear at each step are in A356063.

			36 is divisible by 1, 2, 3, 4, 18, which are all Lucas numbers, and no integer < 36 has 5 divisors that are Lucas numbers, hence a(5) = 36.

Cf. A000032, A304092, A356063.

Similar sequences: A087997 (palindromes), A129655 (Fibonacci), A333456 (Niven).

a(10) from Amiram Eldar, Jul 25 2022

a(11)-a(18) from David A. Corneth, Jul 27 2022

A357172 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly increasing order.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 54, 24, 36, 48, 72, 180, 144, 360, 336, 468, 504, 936, 1008, 1512, 2520, 3024, 5040, 6552, 7560, 22680, 13104, 19656, 49140, 105840, 39312, 78624, 98280, 248976, 334152, 196560, 393120, 668304, 1244880, 1670760, 1867320, 4520880, 3341520, 3734640

Offset: 1

Bernard Schott, Sep 16 2022

This sequence is finite since A009993 is finite with 511 nonzero terms, hence the last term is a(511) = lcm of the 511 nonzero terms of A009993.

a(511) = 8222356410...6120992000 and has 1036 digits. - Michael S. Branicky, Sep 16 2022

			For n=7, the divisors of 54 are {1, 2, 3, 6, 9, 18, 27, 54} of which 7 have their digits in strictly increasing order (all except 54). No integer < 54 has 7 such divisors, so a(7) = 54.

Cf. A009993, A357171, A357173, A160218.

Similar: A087997 (palindromic), A355303 (undulating), A355594 (alternating).

s[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; seq[len_, nmax_] := Module[{v = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n < nmax, i = s[n]; If[i <= len && v[[i]] == 0, v[[i]] = n; c++]; n++]; v]; seq[25, 10^4] (* Amiram Eldar, Sep 16 2022 *)

isok(d) = Set(d=digits(d)) == d; \\ A009993
f(n) = sumdiv(n, d, isok(d)); \\ A357171
a(n) = my(k=1); while (f(k) !=n, k++); k; \\ Michel Marcus, Sep 16 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return s == "".join(sorted(set(s)))
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 37))) # Michael S. Branicky, Sep 16 2022

More terms from Amiram Eldar, Sep 16 2022

A378140 a(n) is the least palindrome that has exactly n palindromic divisors other than itself and 1.

Original entry on oeis.org

1, 4, 6, 232, 44, 636, 66, 484, 888, 616, 2442, 2112, 4224, 6006, 2772, 26862, 23232, 232232, 46464, 297792, 66066, 88088, 222222, 252252, 213312, 21122112, 234432, 606606, 828828, 444444, 279972, 21211212, 666666, 2444442, 2114112, 2578752, 888888, 4228224, 42422424, 23555532, 54999945, 82711728

Offset: 0

Robert Israel, Jan 08 2025

			a(4) = 44 because 44 is a palindrome with exactly 4 palindromic divisors other than itself and 1, namely 2, 4, 11 and 22, and no smaller palindrome works.

Cf. A071276, A071277, A087997.

ispali:= proc(n) rev(n) = n end proc:
g:= proc(x) nops(select(ispali,numtheory:-divisors(x) minus {1,x})) end proc:
F:= proc(m)
   local x1,x2,x3;
   if m::even then
     [seq(seq(rev(x1) + 10^(m/2)*x1, x1 = 10^(m/2-1) .. 10^(m/2)-1))]
   else
     [seq(seq(rev(x1) + 10^((m-1)/2)*x2 + 10^((m+1)/2)*x1,x2=0..9),x1=10^((m-1)/2-1)..10^((m-1)/2)-1)];
   fi
end proc:
N:= 50: # for a(0) .. a(N)
V:= Array(0..N): count:= 0:
for d from 1 while count

cp's OEIS Frontend

A340549 Smallest integer with exactly n divisors that are repunits.

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A355699 a(n) is the smallest number that has exactly n repdigit divisors.

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A355771 a(n) is the smallest integer that has exactly n divisors from A333369.

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A355968 a(n) is the smallest number that has exactly n odious divisors (A000069).

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A356019 a(n) is the smallest number that has exactly n evil divisors (A001969).

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A340796 a(n) is the smallest number with exactly n divisors that are Brazilian.

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A355695 a(n) is the smallest number that has exactly n nonpalindromic divisors (A029742).

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