A102554 -id:A102554 - cp's OEIS frontend

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

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

A102550 Number of distinct prime-factors of n that are bitwise covered by n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0

Offset: 1

Reinhard Zumkeller, Jan 14 2005

nonn

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

Index entries for sequences related to binary expansion of n

Cf. A001221 (omega), A007088, A004676, A080099, A102210.

Cf. A102551, A102553, A102554, A102555.

a[1] = 0; a[k_] := Module[{f=FactorInteger[k][[;; , 1]]}, Count[BitAnd[k, f]-f, 0]];  Array[a,120] (* Amiram Eldar, Feb 06 2019 *)

a(A102553(n)) = A001221(A102553(n));
a(A102554(n)) < A001221(A102554(n));
a(A102551(n)) = 0, a(A102551(n)) > 0;
a(A102555(n)) = n;
a(m) < n for m < A102555(n).
a(n) = Sum_{p|n} (binomial(n,p) mod 2), where p is a prime. - Ridouane Oudra, May 03 2019

Offset 1 from Amiram Eldar, Feb 06 2019

A355670 Numbers k such that A246600(k) < A000005(k).

Original entry on oeis.org

2, 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

Offset: 1

Chai Wah Wu, Dec 19 2022

Numbers k such that bitwise OR(k, d_1, d_2, ... d_m) > k where d_1, ..., d_m are the divisors of k.
Complement of A359080.
First 21 terms coincide with A336376.
A102554 is a subsequence;  this sequence contains 1, 135, 175, 243, 343, 351, 363, ... which are not in A102554.

Cf. A000005, A102553, A102554, A246600, A336375, A359080.

from itertools import count, islice
from operator import ior
from functools import reduce
from sympy import divisors
def A355670_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n|reduce(ior,divisors(n,generator=True))>n,count(max(startvalue,1)))
A355670_list = list(islice(A355670_gen(), 20))

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

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A102550 Number of distinct prime-factors of n that are bitwise covered by n.

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A355670 Numbers k such that A246600(k) < A000005(k).

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Python