A337741 -id:A337741 - cp's OEIS frontend

A093696 Numbers n such that all divisors of n have an odd number of 1's in their binary expansions.

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 22, 26, 28, 31, 32, 37, 38, 41, 44, 47, 49, 52, 56, 59, 61, 62, 64, 67, 73, 74, 76, 79, 82, 88, 91, 94, 97, 98, 103, 104, 107, 109, 112, 118, 121, 122, 124, 127, 128, 131, 133, 134, 137, 143, 146, 148, 151, 152, 157, 158, 164, 167, 173

Offset: 1

Jason Earls, May 16 2004

Subsequence of A000069. - Michel Marcus, Feb 09 2014

Numbers all of whose divisors are odious. - Bernard Schott, Jul 22 2022

			14 is in the sequence because its divisors are [1, 2, 7, 14] and in binary: 1, 10, 111 and 1110, all have an odd number of 1's.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000069, A001969, A227872, A330289, A355968, A355969.
Similar sequences: A062687, A190217, A337741, A337941, A355596.
A000079 is a subsequence.

isA001969 := proc(n)
    if wt(n) mod 2 = 0 then
        true;
    else
        false;
    end if;
end proc:
isA093696 := proc(n)
    for d in numtheory[divisors](n) do
        if isA001969(d) then
            return false;
        end if;
    end do;
    true;
end proc:
for n from 1 to 200 do
    if isA093696(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Feb 13 2014

odiousQ[n_] := OddQ @ DigitCount[n, 2][[1]]; Select[Range[200], AllTrue[ Divisors[#], odiousQ ] &] (* Amiram Eldar, Dec 09 2019 *)

is(n)=fordiv(n,d,if(hammingweight(d)%2==0, return(0))); 1 \\ Charles R Greathouse IV, Mar 29 2013

from sympy import divisors, isprime
def c(n): return bin(n).count("1")&1
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(174) if ok(k)]) # Michael S. Branicky, Jul 24 2022

{n: A356018(n) =0 }. - R. J. Mathar, Aug 07 2022

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

Original entry on oeis.org

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

A355773 Numbers all of whose divisors are members of A333369.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 31, 35, 37, 39, 51, 53, 57, 59, 71, 73, 79, 91, 93, 95, 97, 111, 137, 139, 153, 157, 159, 173, 179, 193, 197, 221, 223, 227, 229, 317, 333, 359, 371, 379, 395, 397, 443, 449, 519, 537, 571, 579, 591, 593, 661, 663, 669, 719, 739

Offset: 1

Bernard Schott, Jul 18 2022

All terms are necessarily odd because 2 is not in A333369

			111 is a term since all the divisors of 111, i.e., 1, 3, 37 and 111, are in A333369.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Project Euler, Problem 520: Simbers.

Cf. A333369, A353735, A355770, A355771, A355772.
Similar sequences: A062687, A190217, A329419, A337741
.
Subsequences: A155045, A355853.

simQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[1000], AllTrue[Divisors[#], simQ] &] (* Amiram Eldar, Jul 19 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
isok(k) = fordiv(k, d, if (!issimber(d), return(0))); return(1); \\ Michel Marcus, Jul 19 2022

from sympy import divisors, isprime
def c(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(740) if ok(k)]) # Michael S. Branicky, Jul 24 2022

A337941 Numbers whose divisors are all Zuckerman numbers (A007602).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 1111111111111111111, 11111111111111111111111

Offset: 1

Bernard Schott, Oct 01 2020

Inspired by A337741.
Zuckerman numbers are numbers that are divisible by the product of their digits (see link).
The next term is the repunit prime R_317 which is too large to include in the data.
Primes in this sequence are 2, 3, 5, 7 and all the repunit primes (see A004023).
This sequence is infinite if and only if there are infinitely many repunit primes.

			6 is a term since all the divisors of 6, i.e., 1, 2, 3 and 6, are Zuckerman numbers.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty

Subsequence of A007602.
Similar sequences: A062687, A190217, A308851, A329419, A337741.
Cf. A004023, A335037, A335038.
Cf. A004022 (subsequence of prime repunits).

zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; Select[Range[24], AllTrue[Divisors[#], zuckQ] &] (* Amiram Eldar, Oct 01 2020 *)

isok(m) = {fordiv(m, d, my(p=vecprod(digits(d))); if (!p || (d % p), return (0))); return (1);} \\ Michel Marcus, Oct 05 2020

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

A337819 a(n) is the smallest number k for which k*d is a Niven number, for any divisor d of n, n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 9, 3, 2, 3, 6, 1, 6, 1, 1, 10, 9, 1, 2, 9, 1, 3, 9, 2, 12, 9, 10, 6, 6, 1, 3, 6, 9, 1, 10, 3, 12, 10, 2, 9, 9, 3, 9, 2, 6, 9, 18, 1, 10, 9, 6, 9, 9, 2, 12, 18, 1, 9, 12, 10, 3, 6, 9, 6, 18, 1, 7, 3, 2, 9, 10, 9, 9, 9, 1, 10

Offset: 1

Marius A. Burtea, Sep 23 2020

a(n) = 1 if and only if n is in A337741.

			The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are in A337741, so a(1) = a(2) = ... = a(9) = a(10) = 1.
For n = 11 the divisors are 1, 11 and 10 * 1 = 10 = A005349(10) and 10 * 11 = 110 = A005349(36), so a(11) = 10.
For n = 14 the divisors are 1, 2, 7, 14 and 3 * 1 = 3 = A005349(3), 3 * 2 = 6 = A005349(6), 3 * 7 = 21 = A005349(14), 3 * 14 = 42 = A005349(20), so a(14) = 3.
For n = 40 , A337741(18) = 40, so a(40) = 1.

Cf. A005349, A144261, A337741.

niven:=func; a:=[]; for n in [1..90] do k:=1; while not forall{d: d in Divisors(n)| niven(k*d)} do k:=k+1; end while; Append(~a,k); end for; a;

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; a[n_] := Module[{k = 1}, While[!AllTrue[k * Divisors[n], nivenQ], k++]; k]; Array[a, 100] (* Amiram Eldar, Sep 23 2020 *)

is(n) = n%sumdigits(n)==0; \\ A005349
isok(n, k) = fordiv(n, d, if (!is(k*d), return(0))); return(1);
a(n) = {my(k=1); while (! isok(n,k), k++); k;} \\ Michel Marcus, Sep 24 2020

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A093696 Numbers n such that all divisors of n have an odd number of 1's in their binary expansions.

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

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A355773 Numbers all of whose divisors are members of A333369.

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A337941 Numbers whose divisors are all Zuckerman numbers (A007602).

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

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A337819 a(n) is the smallest number k for which k*d is a Niven number, for any divisor d of n, n >= 1.

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