A333456 -id:A333456 - cp's OEIS frontend

A332268 a(n) is the number of divisors of n that are Niven numbers.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 4, 2, 1, 8, 2, 2, 4, 4, 1, 7, 1, 4, 2, 2, 3, 9, 1, 2, 2, 8, 1, 7, 1, 3, 5, 2, 1, 9, 2, 5, 2, 3, 1, 8, 2, 5, 2, 2, 1, 11, 1, 2, 6, 4, 2, 4, 1, 3, 2, 6, 1, 12, 1, 2, 3, 3, 2, 4, 1, 9, 5, 2, 1, 10, 2, 2

Offset: 1

Marius A. Burtea, May 04 2020

If p is a prime number, p >= 11, then a(p) = 1.
Numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 36, 40, 54, 63, 72, 81, 108, 162, 216, 243, 324, 486, 648, 972, 1944, have all divisors Niven numbers. There are only finitely many numbers all of whose divisors are Niven numbers. (A337741).
A333456(n) is the least number k such that a(k) = n. - Bernard Schott, Jul 30 2022

			For n = 4 the divisors are 1, 2, 4 and they are all Niven numbers, so a(4) = 3.
For n = 14 the divisors are 1, 2, 7 and 14. Only 1, 2 and 7 are Niven numbers, so a(14) = 3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000005, A005349, A333456, A337741, A340637.

[#[d:d in Divisors(k)|d mod &+Intseq(d) eq 0]:k  in [1..100]];

a[n_] := DivisorSum[n, 1 &, Divisible[#, Plus @@ IntegerDigits[#]] &]; Array[a, 100] (* Amiram Eldar, May 04 2020 *)

a(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ Michel Marcus, May 04 2020

a(A333456(n)) = n. - Bernard Schott, Jul 30 2022

A335038 a(n) is the smallest number m with exactly n divisors that are Zuckerman numbers, or -1 if there is no such m.

Original entry on oeis.org

1, 2, 4, 6, 18, 12, 84, 24, 168, 72, 144, 360, 432, 1080, 2016, 2160, 6048, 8064, 15120, 34272, 24192, 60480, 48384, 88704, 120960, 354816, 241920, 483840, 665280, 266112, 798336, 532224, 1596672, 1064448, 1862784, 2661120, 3725568, 5322240, 10644480, 7451136

Offset: 1

Bernard Schott, Jun 03 2020

Inspired by A333456.
A Zuckerman number (A007602) is a number that is divisible by the product of its digits; e.g., 24 is a Zuckerman number because it is divisible by 2*4=8.
The divisors 1 and m (if m is itself a Zuckerman number) are included.
Conjecture: m always exists.
Not all terms in the sequence are Zuckerman numbers. For example a(7) = 84 has product of digits = 32 and 84/32 = 21/8 = 2.625.

			Of the six divisors of 18, five are Zuckerman numbers: 1, 2, 3, 6 and 9, and there is no smaller number with five Zuckerman divisors, hence a(5) = 18.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Cf. A007602, A335037, A333456 (similar, with Niven divisors).

zuckQ[n_] := (prodig = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prodig]; numDiv[n_] := Count[Divisors[n], ?(zuckQ[#] &)]; mx = 50; n = 1; c = 0; v = Table[0, {mx}]; While[c < mx, i = numDiv[n]; If[i <= mx && v[[i]] == 0, c++; v[[i]] = n]; n++]; v (* _Amiram Eldar, Jun 03 2020 *)

iszu(n) = my(p=vecprod(digits(n))); p && !(n % p);
a(n) = {my(k=1); while (sumdiv(k, d, iszu(d)) !=n, k++); k;} \\ Michel Marcus, Jun 03 2020

More terms from Amiram Eldar, Jun 03 2020

Edited, added escape clause. - N. J. A. Sloane, Jun 04 2020

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

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

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

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A332268 a(n) is the number of divisors of n that are Niven numbers.

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A335038 a(n) is the smallest number m with exactly n divisors that are Zuckerman numbers, or -1 if there is no such m.

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

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