A093642 -id:A093642 - cp's OEIS frontend

A105441 Numbers with at least two odd prime factors (not necessarily distinct).

9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 39, 42, 45, 49, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141

Offset: 1

Reinhard Zumkeller, Apr 09 2005

Also polite numbers (A138591) that can be expressed as the sum of two or more consecutive integers in more than one ways. For example 9=4+5 and 9=2+3+4. Also 15=7+8, 15=4+5+6 and 15=1+2+3+4+5. - Jayanta Basu, Apr 30 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Complement of A093641; A093642 is a subsequence.

Cf. A001227, A087436, A138591.

a105441 n = a105441_list !! (n-1)
a105441_list = filter ((> 2) . a001227) [1..]
-- Reinhard Zumkeller, May 01 2012

opf3Q[n_]:=Count[Flatten[Table[First[#],{Last[#]}]&/@FactorInteger[n]], ?OddQ]>1 (* _Harvey P. Dale, Jun 13 2011 *)

upTo(lim)=my(v=List(),p=7,m);forprime(q=8,lim,forstep(n=p+2,q-2,2,m=n;while(m<=lim,listput(v,m);m<<=1));p=q);forstep(n=p+2,lim,2,listput(v,n));vecsort(Vec(v)) \\ Charles R Greathouse IV, Aug 08 2011

is(n)=n>>=valuation(n,2); !isprime(n) && n>1 \\ Charles R Greathouse IV, Apr 30 2013

from sympy import primepi
def A105441(n):
    def f(x): return int(n+1+sum(primepi(x>>i) for i in range(x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 02 2025

A087436(a(n)) > 1.

A001227(a(n)) > 2. [Reinhard Zumkeller, May 01 2012]

A093640 Number of divisors of n whose binary representation is contained in that of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 07 2004

			n = 18: divisors of 18: 1 = '1', 2 = '10', 3 = '11', 6 = '110', 9 = '1001' and 18 = '10010': four of them are binary substrings of '10010', therefore a(18) = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n.

Cf. A000005, A078822, A007088, A093641, A093642.

import Data.List (isInfixOf)
a093640 n  = length [d | d <- [1..n], mod n d == 0,
                         show (a007088 d) `isInfixOf` show (a007088 n)]
-- Reinhard Zumkeller, Jan 22 2012

a[n_] := DivisorSum[n, 1 &, StringContainsQ @@ IntegerString[{n, #}, 2] &]; Array[a, 100] (* Amiram Eldar, Jul 16 2022 *)

from sympy import divisors
def a(n):
    s = bin(n)[2:]
    return sum(1 for d in divisors(n, generator=True) if bin(d)[2:] in s)
print([a(n) for n in range(1, 102)]) # Michael S. Branicky, Jul 11 2022

a(n) > 1 for n>1.
a(p) = 2 for primes p.
a(A093641(n)) = A000005(A093641(n)).
a(A093642(n)) < A000005(A093642(n)).

A123345 Numbers containing all divisors in their binary representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 55, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118

Offset: 1

Reinhard Zumkeller, Oct 12 2006

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

Complement of A093642. Different from A093641.

Cf. A000040 (subsequence), A027750, A218388.

import Data.List (unfoldr, isInfixOf)
a123345 n = a123345_list !! (n-1)
a123345_list = filter
  (\x -> all (`isInfixOf` b x) $ map b $ a027750_row x) [1..] where
  b = unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, Oct 27 2012

q[n_] := AllTrue[Divisors[n], StringContainsQ[IntegerString[n, 2], IntegerString[#, 2]] &]; Select[Range[100], q] (* Amiram Eldar, Jun 05 2022 *)

from sympy import divisors
def ok(n):
    b = bin(n)[2:]
    return n and all(bin(d)[2:] in b for d in divisors(n, generator=True))
print([k for k in range(119) if ok(k)]) # Michael S. Branicky, Jun 05 2022

A105442 Numbers with at least two odd prime factors (not necessarily distinct) such that in binary representation all divisors of n are contained in n.

Original entry on oeis.org

55, 215, 407, 1403, 1681, 3223, 3362, 3415, 6724, 13448, 13655, 15487, 25751, 80089, 146621, 160178, 218455, 237169, 320356, 445663, 464711, 474338, 873815, 948676, 1662743, 1897352, 1932377, 1975531, 2484187, 3223001, 3410639, 3872639

Offset: 1

Reinhard Zumkeller, Apr 09 2005

nonn

Intersection of A093641 and A105441;

A087436(a(n)) > 1.

Cf. A093642, A093641.

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A105441 Numbers with at least two odd prime factors (not necessarily distinct).

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A093640 Number of divisors of n whose binary representation is contained in that of n.

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A123345 Numbers containing all divisors in their binary representation.

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Haskell

Mathematica

Python

A105442 Numbers with at least two odd prime factors (not necessarily distinct) such that in binary representation all divisors of n are contained in n.

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Author

Keywords

Comments

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