A340548 -id:A340548 - cp's OEIS frontend

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

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

A350756 Integers whose number of divisors that are triangular numbers sets a new record.

Original entry on oeis.org

1, 3, 6, 30, 90, 180, 210, 420, 630, 1260, 2520, 6930, 13860, 27720, 41580, 83160, 138600, 180180, 360360, 540540, 1081080, 1413720, 2162160, 3063060, 6126120, 12252240, 18378360, 36756720, 73513440, 91891800, 116396280, 183783600, 232792560, 349188840

Offset: 1

Bernard Schott, Jan 13 2022

nonn

Terms that are triangular: 1, 3, 6, 210, 630, 2162160, ...

The number of triangular divisors of a(n) is A007862(a(n)): 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, ...

			1260 has 36 divisors of which 12 are triangular numbers {1, 3, 6, 10, 15, 21, 28, 36, 45, 105, 210, 630}. No positive integer smaller than 1260 has as many as twelve triangular divisors; hence 1260 is a term.

Cf. A000217, A007862, A130317.

Similar for A046952 (squares), A053624 (odd), A093036 (palindromes), A181808 (even), A340548 (repdigits), A340549 (repunits) divisors.

max=0;Do[If[(d=Length@Select[Divisors@k,IntegerQ[(Sqrt[8#+1]-1)/2]&])>max,Print@k;max=d],{k,10^10}] (* Giorgos Kalogeropoulos, Jan 13 2022 *)

lista(nn) = {my(r=0); for (n=1, nn, my(m = sumdiv(n, d, ispolygonal(d,3))); if (m>r, r=m; print1(n", ")));} \\ Michel Marcus, Jan 14 2022

A355304 Integers whose number of normal undulating divisors sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 1080, 1260, 1440, 1680, 2160, 2520, 5040, 7560, 10080, 15120, 21840, 28080, 32760, 56160, 65520, 98280, 131040, 196560, 393120, 589680, 786240, 1113840, 1670760, 2227680, 3341520, 6683040, 13366080, 20049120

Offset: 1

Bernard Schott, Jun 30 2022

Normal undulating integers are in A355301.
The first 14 terms are also the first 14 highly composite numbers in A002182, then A002182(15) = 840 while a(15) = 1080. Indeed, 840 is the smallest integer that has 32 divisors of which only 28 are normal undulating integers, while 1080 has also 32 divisors of which 30 are normal undulating integers.
Corresponding records of number of normal undulating divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12, ...

			a(6) = 24 is in the sequence because A355302(24) is larger than any earlier value in A355302.

Cf. A002182, A355301, A355302, A355303.

Similar, but with divisors that are: A046952 (squares), A053624 (odd), A181808 (even), A093036 (palindromes), A340548 (repdigits), A340549 (repunits), A350756 (triangular).

nuQ[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; dm = -1; seq = {}; Do[If[(d = DivisorSum[n, 1 &, nuQ[#] &]) > dm, dm = d; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jun 30 2022 *)

More terms from Amiram Eldar, Jun 30 2022

A340637 Integers whose number of divisors that are Niven numbers sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 720, 1080, 1800, 2160, 2520, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 30240, 45360, 50400, 60480, 75600, 90720, 100800, 110880, 120960, 151200, 166320, 221760, 277200, 302400, 332640, 453600, 498960, 554400

Offset: 1

Bernard Schott, Jan 14 2021

A Niven number (A005349) is a number that is divisible by the sum of its digits.

The first 13 terms are the first 13 terms of A236021, then A236021(14) = 420 while a(14) = 720.

			The 8 divisors of 24 are all Niven numbers, and also, 24 is the smallest integer that has at least 8 divisors that are Niven numbers, hence 24 is a term.

Cf. A005349, A236021, A332268, A337741.
Subsequence of A333456.
Similar for palindromes (A093036), repdigits (A340548), repunits (A340549), Zuckerman numbers (A340638).

s[n_] := DivisorSum[n, 1 &, Divisible[#, Plus @@ IntegerDigits[#]] &]; smax = 0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^6}]; seq (* Amiram Eldar, Jan 14 2021 *)

f(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ A332268
lista(nn) = {my(m=0); for (n=1, nn, my(x = f(n)); if (x > m, m = x; print1(n, ", ")););} \\ Michel Marcus, Jan 14 2021

More terms from Amiram Eldar, Jan 14 2021

A340797 Integers whose number of divisors that are Brazilian sets a new record.

Original entry on oeis.org

1, 7, 14, 24, 40, 48, 60, 84, 120, 168, 240, 336, 360, 420, 672, 720, 840, 1260, 1680, 2520, 3360, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 43680, 45360, 50400, 55440, 65520, 83160, 98280, 110880, 131040, 166320, 196560, 221760, 262080, 277200, 327600

Offset: 1

Bernard Schott, Jan 24 2021

Corresponding number of Brazilian divisors: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 14, 15, 17, 18, 19, 26, ...

Observation: the 58 consecutive highly composite numbers from A002182(12) = 240 to A002182(69) = 2095133040 (maybe more, according to conjectured terms) are also terms of this sequence.

			40 has 8 divisors: {1, 2, 4, 5, 8, 10, 20, 40} of which 4 are Brazilian: {8, 10, 20, 40}. No positive integer smaller than 40 has as many as four Brazilian divisors; hence 40 is a term.

David A. Corneth, Table of n, a(n) for n = 1..85
David A. Corneth, Conjectured terms
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A002182, A125134, A220570, A308851, A340795, A340796.

Similar with: A093036 (palindromes), A340548 (repdigits), A340549 (repunits), A340637 (Niven), A340638 (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[#] &]; dm = -1; s = {}; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 1000}]; s (* Amiram Eldar, Jan 24 2021 *)

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
nbd(n) = sumdiv(n, d, isb(d)); \\ A340795
lista(nn) = {my(m=-1); for (n=1, nn, my(x = nbd(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 24 2021

a(20)-a(36) from Michel Marcus, Jan 24 2021

a(37)-a(44) from Amiram Eldar, Jan 24 2021

A357173 Positions of records in A357171, i.e., integers whose number of divisors whose decimal digits are in strictly increasing order sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 72, 144, 336, 468, 504, 936, 1008, 1512, 2520, 3024, 5040, 6552, 7560, 13104, 19656, 39312, 78624, 98280, 196560, 393120, 668304, 1244880, 1670760, 1867320, 3341520, 3734640, 7469280, 22407840, 26142480, 31744440, 52284960, 63488880

Offset: 1

Bernard Schott, Sep 17 2022

As A009993 is finite, this sequence is necessarily finite.

Corresponding records are 1, 2, 3, 4, 6, 8, 9, 10, 11, ...

			a(6) = 24 is in the sequence because A357171(24) = 8 is larger than any earlier value in A357171.

Cf. A009993, A357171, A357172, A160218.

Similar sequences: A093036, A340548, A355595.

s[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; seq = {}; sm = 0; Do[If[(sn = s[n]) > sm, sm = sn; AppendTo[seq, n]], {n, 1, 10^4}]; seq (* Amiram Eldar, Sep 17 2022 *)

isok(d) = Set(d=digits(d)) == d; \\ A009993
f(n) = sumdiv(n, d, isok(d)); \\ A357171
lista(nn) = my(r=0, list = List()); for (k=1, nn, my(m=f(k)); if (m>r, listput(list, k); r = m);); Vec(list); \\ Michel Marcus, Sep 18 2022

More terms from Amiram Eldar, Sep 17 2022

A340638 Integers whose number of divisors that are Zuckerman numbers sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 72, 144, 360, 432, 1080, 2016, 2160, 6048, 8064, 15120, 24192, 48384, 88704, 120960, 241920, 266112, 532224, 1064448, 1862784, 2661120, 3725568, 5322240, 7451136, 10450944, 19160064, 20901888, 28740096, 38320128, 57480192, 99283968, 114960384

Offset: 1

Bernard Schott, Jan 14 2021

A Zuckerman number is a number that is divisible by the product of its digits (A007602).
The terms in this sequence are not necessarily Zuckerman numbers. For example a(7) = 72 has product of digits = 14 and 72/14 = 36/7 = 5.142...
The first seven terms are the first seven terms of A087997, then A087997(8) = 66 while a(8) = 144.

			The 8 divisors of 24 are all Zuckerman numbers, and also, 24 is the smallest integer that has at least 8 divisors that are Zuckerman numbers, hence 24 is a term.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Cf. A007602, A335037, A337941.
Subsequence of A335038.
Similar for palindromes (A093036), repdigits (A340548), repunits (A340549), Niven numbers (A340637).

zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; s[n_] := DivisorSum[n, 1 &, zuckQ[#] &]; smax = 0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jan 14 2021 *)

isokz(n) = iferr(!(n % vecprod(digits(n))), E, 0); \\ A007602
lista(nn) = {my(m=0); for (n=1, nn, my(x = sumdiv(n, d, isokz(d));); if (x > m, m = x; print1(n, ", ")););} \\ Michel Marcus, Jan 15 2021

More terms from David A. Corneth and Amiram Eldar, Jan 15 2021

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

A358101 Positions of records in A358099, i.e., integers whose number of divisors whose decimal digits are in strictly decreasing order sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 30, 40, 60, 120, 240, 360, 420, 840, 1260, 2520, 5040, 8640, 10080, 15120, 20160, 30240, 60480, 120960, 181440, 362880, 544320, 786240, 1572480, 1874880, 3749760, 5624640, 7862400, 14938560, 23587200, 24373440, 31872960, 63745920, 95618880

Offset: 1

Bernard Schott, Nov 03 2022

As A009995 is finite, this sequence is necessarily finite.

Corresponding records are 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, ...

			a(9) = 60 is in the sequence because A358099(60) = 10 is larger than any earlier value in A358099.

Cf. A009995, A190219, A358099, A358100.

Similar sequences: A093036, A340548, A357173.

f[n_] := DivisorSum[n, 1 &, Greater @@ IntegerDigits[#] &]; fm = 0; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Nov 03 2022 *)

More terms from Amiram Eldar, Nov 03 2022

cp's OEIS Frontend

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

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A350756 Integers whose number of divisors that are triangular numbers sets a new record.

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A355304 Integers whose number of normal undulating divisors sets a new record.

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A340637 Integers whose number of divisors that are Niven numbers sets a new record.

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A340797 Integers whose number of divisors that are Brazilian sets a new record.

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A357173 Positions of records in A357171, i.e., integers whose number of divisors whose decimal digits are in strictly increasing order sets a new record.

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A340638 Integers whose number of divisors that are Zuckerman numbers sets a new record.

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