A080943 -id:A080943 - cp's OEIS frontend

A080942 Number of divisors of n that are also suffixes of n in binary representation.

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

Offset: 1

Reinhard Zumkeller, Feb 25 2003

a(n) = 1 iff n = 2^k (A000079), the only divisor is n itself.

For a(n) > 1 the other trivial divisor is 1 for odd numbers and 2 for even numbers (A057716).

			n=63 has A000005(63)=6 divisors: 1='1', 3='11', 7='111', 9='1001', 21='10101' and 63='111111', {1,11,111,111111} are also suffixes of 111111, therefore a(63)=4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007088, A080940, A080941, A080943, A080945, A080946, A080947, A080948.

Cf. A000005, A027750, A030308, A057716, A239826.

import Data.List (isPrefixOf); import Data.Function (on)
a080942 n = length $
            filter ((flip isPrefixOf `on` a030308_row) n) $ a027750_row n
-- Reinhard Zumkeller, Mar 27 2014

a[n_] := DivisorSum[n, 1 &, Mod[n, 2^BitLength[#]] == # &]; Array[a, 100] (* Amiram Eldar, Apr 07 2023 *)

from sympy import divisors
def A080942(n): return sum(1 for d in divisors(n,generator=True) if not (d^n)&((1<Chai Wah Wu, Jun 20 2023

a(A080943(n)) = 2.
a(A080945(n)) > 2.
a(A080946(n)) = 3.
a(A080947(n)) > 3.
a(n) <= A000005(n).
a(p) = 2 for odd primes p.
a(A080948(n)) = n and a(m) < n for m < A080948(n).

A080945 Numbers having more than two divisors that are also suffixes in binary representation.

Original entry on oeis.org

15, 27, 30, 39, 45, 51, 54, 60, 63, 75, 78, 85, 87, 90, 99, 102, 108, 111, 119, 120, 123, 125, 126, 135, 147, 150, 153, 156, 159, 165, 170, 171, 174, 175, 180, 183, 187, 195, 198, 204, 205, 207, 216, 219, 221, 222, 231, 238, 240, 243, 245, 246, 250, 252, 255

Offset: 1

Reinhard Zumkeller, Feb 25 2003

A080942(a(n))>2; complement of A080944;

A080942(a(n))-2 < A070824(a(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A080943, A080946, A007088, A080940, A080941.

a080945 n = a080945_list !! (n-1)
a080945_list = filter ((> 2) . a080942) [1..]
-- Reinhard Zumkeller, Mar 27 2014

A080946 Numbers having exactly three divisors that are also suffixes in binary representation.

Original entry on oeis.org

15, 27, 30, 39, 45, 51, 54, 60, 75, 78, 85, 87, 90, 99, 102, 108, 111, 119, 120, 123, 125, 135, 147, 150, 153, 156, 159, 165, 170, 171, 174, 175, 180, 183, 187, 195, 198, 204, 205, 207, 216, 219, 221, 222, 238, 240, 243, 245, 246, 250, 267, 270, 279, 285, 287

Offset: 1

Reinhard Zumkeller, Feb 25 2003

A080942(a(n))=3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A080943, A080945, A080947, A007088, A080940, A080941.

a080946 n = a080946_list !! (n-1)
a080946_list = filter ((== 3) . a080942) [1..]
-- Reinhard Zumkeller, Mar 27 2014

A080944 Numbers having only trivial divisors that are also suffixes in binary representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Reinhard Zumkeller, Feb 25 2003

A080942(a(n))<=2; union of powers of 2 (A000079) and A080943; complement of A080945.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007088, A080940, A080941.

a080944 n = a080944_list !! (n-1)
a080944_list = filter ((<= 2) . a080942) [1..]
-- Reinhard Zumkeller, Mar 27 2014

A363690 Numbers k such that A246600(k) = 2.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 84, 86, 88

Offset: 1

Amiram Eldar, Jun 16 2023

A subsequence of A080943 and first differs from it at n = 42: A080943(42) = 55 is not a term of this sequence.
Numbers k such that A246600(k) = 1 are the powers of 2 (A000079).
Numbers k that have exactly 2 divisors d such that the bitwise AND of k and d is equal to d, or equivalently, the bitwise OR of k and d is equal to k. These two divisors are k and the highest power of 2 dividing k, A006519(k).
Includes all the even squarefree semiprimes (i.e., the odd primes doubled, A100484 \ {4}).
If k is a term then 2*k is also a term. The primitive terms are the odd terms of this sequence, A363691.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 76, 681, 6268, 60002, 587247, 5811449, 57817051, 576781821, 5761341533, 57583082392, 575687822743, ... . Apparently, the asymptotic density of this sequence exists and equals 0.575... .

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

Cf. A000079, A006519, A080943, A100484, A246600, A363691.

q[n_] := DivisorSum[n, Boole[BitOr[#, n] == n] &] == 2; Select[Range[100], q]

is(n) = sumdiv(n, d, bitor(d, n) == n) == 2;

from itertools import count, islice
from sympy import divisors
def A363690_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(m:=n&-n)!=n and all(d==m or d==n or n|d!=n for d in divisors(n,generator=True)),count(max(startvalue,2)))
A363690_list = list(islice(A363690_gen(),20)) # Chai Wah Wu, Jun 20 2023

A080948 Least number m having n divisors that are also suffixes of m in binary representation.

Original entry on oeis.org

1, 3, 15, 63, 735, 4095, 185535, 5810175, 1277603775

Offset: 1

Reinhard Zumkeller, Feb 25 2003

a(n) = Min{k: A080942(k)=n}; A080942(a(n)) = n.

2^32 < a(10) <= 12754994175. a(11) <= 5301966462975. - Donovan Johnson, Oct 28 2010

			a(1)=A000079(0)=1; a(2)=A080943(1)=3; a(3)=A080946(1)=15.

Cf. A080940, A080941, A007088.

a(7)-a(9) from Donovan Johnson, Oct 28 2010

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A080942 Number of divisors of n that are also suffixes of n in binary representation.

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A080945 Numbers having more than two divisors that are also suffixes in binary representation.

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A080946 Numbers having exactly three divisors that are also suffixes in binary representation.

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A080944 Numbers having only trivial divisors that are also suffixes in binary representation.

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A363690 Numbers k such that A246600(k) = 2.

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A080948 Least number m having n divisors that are also suffixes of m in binary representation.

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