A080945 -id:A080945 - 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).

A080943 Numbers having exactly two divisors that are also suffixes in binary representation.

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, 55, 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, 89, 91, 92

Offset: 1

Reinhard Zumkeller, Feb 25 2003

A080942(a(n))=2, the two divisors are n and 1 for odd numbers and 2 for even numbers.

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

Cf. A080945, A080944, A007088, A080940, A080941.

a080943 n = a080943_list !! (n-1)
a080943_list = filter ((== 2) . a080942) [1..]
-- Reinhard Zumkeller, Mar 27 2014

from itertools import count, islice
from sympy import divisors
def A080943_gen(startvalue=3): # generator of terms >= startvalue
    return filter(lambda n:(m:=n&-n)!=n and all(d==m or d==n or (d^n)&((1<A080943_list = list(islice(A080943_gen(),20)) # Chai Wah Wu, Jun 20 2023

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

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

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A080943 Numbers having exactly 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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