A363691 -id:A363691 - cp's OEIS frontend

A363693 Terms of A363691 with a record number of divisors.

3, 9, 21, 81, 105, 225, 945, 5265, 5985, 11025, 16065, 36225, 89505, 105105, 187425, 345345, 389025, 1044225, 2027025, 4189185, 6185025, 20307105, 27776385, 76039425, 107972865, 286711425, 402026625, 1853445825, 2440353825, 3807428625, 5106886785, 9449834625

Offset: 1

Amiram Eldar, Jun 16 2023

Odd numbers k with a record number of divisors such that for all the nontrivial divisors d of k (i.e., divisors that are not 1 or k) the bitwise AND of k and d is not equal to d, or equivalently, the bitwise OR of k and d is not equal to k.

The corresponding record values are 2, 3, 4, 5, 8, 9, 16, 20, 24, 27, 32, 36, 40, 48, ... .

Cf. A000005, A359082, A359083, A361937, A363691, A363692.

seq[kmax_] := Module[{s = {}, dm = 0, d1}, Do[d1 = DivisorSigma[0, k]; If[d1 > dm && DivisorSum[k, Boole[BitOr[#, k] == k] &] == 2, dm = d1; AppendTo[s, k]], {k, 1, kmax, 2}]; s]; seq[10^5]

lista(kmax) = {my(dm = 0, d1); forstep(k = 1, kmax, 2, d1 = numdiv(k); if(d1 > dm && sumdiv(k, d, bitor(d, k) == k) == 2, dm = d1; print1(k, ", "))); }

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

cp's OEIS Frontend

A363693 Terms of A363691 with a record number of divisors.

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