A102550 -id:A102550 - cp's OEIS frontend

A102210 Number of primes that are bitwise covered by n.

0, 1, 2, 0, 1, 1, 4, 0, 0, 1, 3, 0, 2, 1, 6, 0, 1, 1, 4, 0, 2, 1, 7, 0, 1, 1, 5, 0, 4, 1, 11, 0, 0, 1, 2, 0, 2, 1, 5, 0, 1, 1, 5, 0, 4, 1, 10, 0, 1, 1, 4, 0, 4, 1, 9, 0, 2, 1, 8, 0, 8, 1, 18, 0, 0, 1, 3, 0, 1, 1, 6, 0, 1, 1, 5, 0, 3, 1, 10, 0, 1, 1, 6, 0, 2, 1, 10, 0, 3, 1, 9, 0, 6, 1, 17, 0, 1, 1, 4, 0, 4, 1

Offset: 1

Reinhard Zumkeller, Dec 30 2004

p is bitwise covered by n iff (p = (n AND p)) bitwise: A080099(n,p)=p.

			n=21->10101 -> a(21) = #{00101=5,10001=17} = 2.

Cf. A007053, A007088, A004676, A080099, A102211, A102212, A102213, A102550.

[#[p:p in PrimesUpTo(n)| p eq BitwiseAnd(n,p)] :n in [1..105] ]; // Marius A. Burtea, Jan 12 2020

a[n_] := Count[Range[n], ?(PrimeQ[#] && BitAnd[n, #] == # &)]; Array[a, 100] (* _Amiram Eldar, Jan 12 2020 *)

a(A102211(n)) = 0; a(A102212(n)) = 1; a(A102213(n)) > 1.

a(2^k-1) = A007053(k) for k > 1. - Amiram Eldar, Jan 12 2020

A102553 Numbers k such that for all prime-factors p: p = (k AND p), bitwise.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 63, 67, 71, 73, 79, 83, 85, 89, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 125, 127, 131, 135, 137, 139, 143, 149, 151, 157, 163, 167, 173, 175, 179, 181, 187, 191, 193, 197, 199

Offset: 1

Reinhard Zumkeller, Jan 14 2005

Numbers k such that A102550(k) = A001221(k).
Apart from first term, subsequence of A102552;
A000040 is a subsequence.
Numbers k such that the bitwise OR of k with all prime divisors of k is equal to k. - Chai Wah Wu, Dec 18 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A001221, A102550, A102552, A102554, A007088, A004676.

okQ[n_] := AllTrue[FactorInteger[n][[All, 1]], # == BitAnd[n, #]&];
Select[Range[200], okQ] (* Jean-François Alcover, Nov 16 2021 *)

from itertools import count, islice
from operator import ior
from functools import reduce
from sympy import primefactors
def A102553_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n == 1 or n|reduce(ior,primefactors(n))==n,count(max(startvalue,1)))
A102553_list = list(islice(A102553_gen(),20)) # Chai Wah Wu, Dec 18 2022

A102554 Numbers k such that p <> (k AND p) for at least one prime-factor p.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 99, 100, 102, 104

Offset: 1

Reinhard Zumkeller, Jan 14 2005

Numbers k such that A102550(k) < A001221(k).

Numbers k such that the bitwise OR of k and all prime factors of k is not equal to k. - Chai Wah Wu, Dec 18 2022

Cf. A001221, A102550, A102553, A007088, A004676.

isA102554 := proc(n)
    local p;
    for p in numtheory[factorset](n) do
        if p <> ANDnos(p,n) then
            return true
        end if;
    end do:
    false ;
end proc:
for n from 1 to 500 do
    if isA102554(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 20 2023

from itertools import count, islice
from functools import reduce
from operator import ior
from sympy import primefactors
def A102554_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:n|reduce(ior,primefactors(n))!=n,count(max(startvalue,2)))
A102554_list = list(islice(A102554_gen(),20)) # Chai Wah Wu, Dec 18 2022

A102555 Smallest number covering bitwise exactly n prime factors.

Original entry on oeis.org

1, 2, 15, 255, 3135, 41055, 440895, 10705695, 242777535, 4360010655

Offset: 0

Reinhard Zumkeller, Jan 14 2005

A102550(a(n))=n and A102550(m) < n for m < a(n).

a(10) <= 287901348735. - Amiram Eldar, Feb 04 2019

Cf. A007088, A004676, A102550.

npfQ[k_,n_] := Module[{f=FactorInteger[k][[;;,1]]}, Length[f] == n && Count[ BitAnd[k, f] - f, 0] == n]; s={1}; Do[k=2; While[!npfQ[k,n], k++]; AppendTo[s, k],{n, 1, 7}]; s (* Amiram Eldar, Feb 04 2019 *)

Two more terms and "base" keyword from Max Alekseyev, Sep 13 2009

Offset 0 and a(7)-a(9) from Amiram Eldar, Feb 04 2019

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A102210 Number of primes that are bitwise covered by n.

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A102553 Numbers k such that for all prime-factors p: p = (k AND p), bitwise.

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A102554 Numbers k such that p <> (k AND p) for at least one prime-factor p.

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A102555 Smallest number covering bitwise exactly n prime factors.

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