A246600 -id:A246600 - cp's OEIS frontend

A359082 Indices of records in A246600.

1, 3, 15, 63, 255, 495, 4095, 96255, 98175, 130815, 203775, 1048575, 5810175, 6455295, 16777215, 67096575, 88062975, 389656575, 553517055, 850917375, 1157349375, 9141354495, 12826279935, 22828220415, 26818379775, 31684427775, 68719476735, 242870910975, 1168231038975

Offset: 1

Amiram Eldar, Dec 15 2022

Numbers k with a record number of divisors d such that the bitwise OR of k and d is equal to k (or equivalently, the bitwise AND of k and d is equal to d).
All the terms are odd since A246600(2*k) = A246600(k).
This sequence is infinite since A246600(2^m-1) = A000005(2^m-1) = A046801(m), and A046801 is unbounded (A046801(2^(m+1)) > A046801(2^m) for all m >= 0).
The corresponding record values are 1, 2, 4, 6, 8, 11, 24, 25, 28, 32, 35, 48, 56, 89, 96, 105, 121, 127, 148, 162, 216, 243, 245, 256, 319, 358, 512, 633, 768, ... .
2*10^11 < a(28) <= 2^48 - 1.

Cf. A046801, A246600, A359080, A359081, A359083.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; seq={}; sm = 0; Do[If[(sn = s[n]) > sm, sm = sn; AppendTo[seq, n]], {n, 1, 10^6}]; seq

lista(nmax) = {my(list = List(), ndmax = 0, d, s); for(n = 1, nmax, nd = sumdiv(n, d, bitand(d, n)==d); if(nd > ndmax, ndmax = nd; listput(list, n))); Vec(list)};

a(28)-a(29) from Martin Ehrenstein, Dec 19 2022

A359080 Numbers k such that A246600(k) = A000005(k).

Original entry on oeis.org

1, 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, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, 197, 199, 211, 219

Offset: 1

Amiram Eldar, Dec 15 2022

Numbers k such that for all the divisors d of k the bitwise OR of k and d is equal to k (or equivalently, the bitwise AND of k and d is equal to d).
Subsequence of A102553. Terms of A102553 that are not in this sequence: 2, 135, 175, 243, 343, ... .
All the terms are odd since if k is even and d = 1 then bitor(k, 1) > k and thus A246600(k) < A000005(k).
All the odd primes are terms.
All the numbers of the form 2^k-1 (A000225) are terms.
Numbers k such that the bitwise OR(k, d_1, d_2, ..., d_m) = k, where d_1, ..., d_m are the divisors of k. - Chai Wah Wu, Dec 18 2022

Subsequence of A102553.
Subsequences: A000225, A065091.
Cf. A000005, A246600, A359081, A359082, A359083.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; Select[Range[250], s[#] == DivisorSigma[0, #] &]

is(n) = sumdiv(n, d, bitor(d, n) == n) == numdiv(n);

from itertools import count, islice
from operator import ior
from functools import reduce
from sympy import divisors
def A359080_gen(startvalue=1):  # generator of terms >= startvalue
    return filter(
        lambda n: n | reduce(ior, divisors(n, generator=True)) == n,
        count(max(startvalue, 1)),
    )
A359080_list = list(islice(A359080_gen(), 20))  # Chai Wah Wu, Dec 18 2022
print(A359080_list)

A359083 Numbers k such that A246600(k) = A000005(k) and A000005(k) sets a new record.

Original entry on oeis.org

1, 3, 15, 63, 255, 891, 4095, 262143, 1048575, 16777215, 68719476735

Offset: 1

Amiram Eldar, Dec 15 2022

Numbers k with a record number of divisors, such that for all the divisors d of k the bitwise OR of k and d is equal to k (or equivalently, the bitwise AND of k and d is equal to d).
All the terms are odd since all the terms of A359080 are odd.
The corresponding numbers of divisors are 1, 2, 4, 6, 8, 10, 24, 32, 48, 96, 512, ... .
a(12) > 3*10^11, if it exists.

Subsequence of A359080.

Cf. A000005, A246600, A359081, A359082.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; seq={}; dm = 0; Do[d = DivisorSigma[0, n]; If[d > dm && d == s[n], dm = d; AppendTo[seq, n]], {n, 1, 2*10^7}]; seq

lista(nmax) = {my(list = List(), ndmax = 0, d, s); for(n = 1, nmax, nd = numdiv(n); if(nd > ndmax && sumdiv(n, d, bitand(d, n)==d) == nd, ndmax = nd; listput(list, n))); Vec(list)};

A359081 a(n) is the least number k such that A246600(k) = n, and -1 if no such k exists.

Original entry on oeis.org

1, 3, 39, 15, 175, 63, 1275, 255, 1215, 891, 495, 6975, 14175, 26367, 13311, 8127, 20475, 42735, 95931, 69615, 36855, 24255, 404415, 4095, 96255, 423423, 253935, 98175, 913275, 165375, 507375, 130815, 3198975, 1576575, 203775, 2154495, 4398975, 1616895, 1556415

Offset: 1

Amiram Eldar, Dec 15 2022

All the terms are odd since A246600(2*k) = A246600(k).

Cf. A000005, A046801, A246600, A359080, A359082, A359083.

seq[nmax_, kmax_] := Module[{s = Table[0, {nmax}], c = 0, k = 1, i}, While[c < nmax && k < kmax, i = DivisorSum[k, 1 &, BitOr[#, k] == k &]; If[i <= nmax && s[[i]] == 0, c++; s[[i]] = k]; k++]; s]; seq[20, 5*10^6]

lista(nmax, kmax=oo) = {my(s = vector(nmax), c = 0, k = 1, i); while(c < nmax && k < kmax, i = sumdiv(k, d, bitor(d, k) == k); if(i <= nmax && s[i] == 0, c++; s[i] = k); k++); s};

A361937 Numbers k with record values of the ratio A000005(k)/A246600(k) between the total number of divisors of k and the number of divisors d of k such that the bitwise OR of k and d is equal to k.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 336, 420, 840, 1680, 3360, 6720, 7560, 15120, 30240, 60480, 95760, 120960, 176400, 191520, 257040, 352800, 383040, 514080, 1028160, 1681680, 2056320, 2998800, 3112200, 5525520, 5997600, 6224400, 8353800, 12448800, 16216200, 24897600

Offset: 1

Amiram Eldar, Mar 31 2023

This sequence is infinite since the ratio A000005(k)/A246600(k) is unbounded. For example, if k = 2^m then A000005(k)/A246600(k) = m+1.

All the terms except for 1 are in A355670.

			The ratios A000005(k)/A246600(k) for k = 1, 2, 3 and 4 are 1, 2, 1 and 3. The record values, 1, 2 and 3, occur at 1, 2 and 4, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..52

Cf. A000005, A246600, A355670, A359080, A359082, A359083.

Similar sequences: A307870, A335832, A361807.

r[n_] := DivisorSigma[0,n]/DivisorSum[n, Boole[BitOr[#, n] == n] &];
seq[kmax_] := Module[{rm = 0, k = 1, s = {}, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1 , kmax}]; s]; seq[10^6]

r(n) = numdiv(n)/sumdiv(n, d, bitor(d, n) == n);
lista(kmax) = {my(rm = 0, r1); for(k = 1, kmax, r1 = r(k); if(r1 > rm, rm = r1; 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

A363691 Odd numbers k such that A246600(k) = 2.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 89, 91, 93, 97, 101, 103, 105, 107, 109, 113, 115, 117, 121, 127, 129, 131, 133, 137, 139, 141, 145, 149, 151, 155, 157, 161, 163, 167

Offset: 1

Amiram Eldar, Jun 16 2023

Odd 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 1 and k.
The terms of this sequence are the primitive terms of A363690: If m is a term, then 2^k*m is a term of A363690 for all k >= 0.
Includes all the odd primes (A065091), and all the squares of odd primes (A001248 \ {4}).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001248, A065091, A246600, A363690.

q[n_] := DivisorSum[n, Boole[BitOr[#, n] == n] &] == 2; Select[Range[1, 200, 2], q]

is(n) = n % 2 && sumdiv(n, d, bitor(d, n) == n) == 2;

from itertools import count, islice
from sympy import divisors
def A363691_gen(startvalue=3): # generator of terms >= startvalue
    return filter(lambda n:all(d==1 or d==n or n|d!=n for d in divisors(n,generator=True)),count(max(startvalue,3)|1,2))
A363691_list = list(islice(A363691_gen(),20)) # Chai Wah Wu, Jun 20 2023

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))

A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 30 2022

First differs from A318672 and A325989 at n = 32.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both an infinitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e and the bitwise AND of e and k is equal to k.
The least term that is higher than 2 is a(216) = 4.
The position of the first appearance of a prime p in this sequence is 2^A359081(p), if A359081(p) > -1. E.g., 2^39 = 549755813888 for p = 3, 2^175 = 4.789...*10^52 for p = 5, and 2^1275 = 6.504...*10^383 for p = 7.
This sequence is unbounded since A246600 is unbounded (see A359082).

			a(8) = 2 since 8 has 2 divisors that are both infinitary and exponential: 2 and 8.

Cf. A037445, A049419, A077609, A080948, A138302, A246600, A322791, A359081, A359082, A359412.

Cf. A318672, A325989.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = sumdiv(n, d, bitand(d, n)==d);
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i,2]));}

from math import prod
from sympy import divisors, factorint
def A359411(n): return prod(sum(1 for d in divisors(e,generator=True) if e|d == e) for e in factorint(n).values()) # Chai Wah Wu, Sep 01 2023

Multiplicative with a(p^e) = A246600(e).
a(n) = 1 if and only if n is in A138302.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A246600(k)/p^k) = 1.135514937... .

A246601 Sum of divisors d of n with property that the binary representation of d can be obtained from the binary representation of n by changing any number of 1's to 0's.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 8, 8, 10, 12, 12, 16, 14, 16, 24, 16, 18, 20, 20, 24, 22, 24, 24, 32, 26, 28, 40, 32, 30, 48, 32, 32, 34, 36, 36, 40, 38, 40, 43, 48, 42, 44, 44, 48, 60, 48, 48, 64, 50, 52, 72, 56, 54, 80, 61, 64, 58, 60, 60, 96, 62, 64, 104, 64, 66, 68, 68, 72, 70, 72

Offset: 1

N. J. A. Sloane, Sep 06 2014

Equivalently, the sum of the divisors d of n such that the bitwise OR of d and n is equal to n. - Chai Wah Wu, Sep 06 2014

Equivalently, the sum of the divisors d of n such that the bitwise AND of n and d is equal to d. - Amiram Eldar, Dec 15 2022

			12 = 1100_2; only the divisors 4 = 0100_2 and 12 = 1100_2 satisfy the condition, so(12) = 4+12 = 16.
15 = 1111_2; all divisors 1,3,5,15 satisfy the condition, so a(15)=24.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A075708, A246600.

with(numtheory);
sd:=proc(n) local a,d,s,t,i,sw;
s:=convert(n,base,2);
a:=0;
for d in divisors(n) do
sw:=-1;
t:=convert(d,base,2);
for i from 1 to nops(t) do if t[i]>s[i] then sw:=1; fi; od:
if sw<0 then a:=a+d; fi;
od;
a;
end;
[seq(sd(n),n=1..100)];

a[n_] := DivisorSum[n, #*Boole[BitOr[#, n] == n] &]; Array[a, 100] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)

a(n) = sumdiv(n, d, d*(bitor(n,d)==n)); \\ Michel Marcus, Sep 07 2014

from sympy import divisors
def A246601(n):
    return sum(d for d in divisors(n) if n|d == n)
# Chai Wah Wu, Sep 06 2014

a(2^i) = 2^i.
a(odd prime p) = p+1.
From Amiram Eldar, Dec 15 2022: (Start)
a(2*n) = 2*a(n), and therefore a(m*2^k) = 2^k*a(m) for m odd and k>=0.
a(2^n-1) = sigma(2^n-1) = A075708(n). (End)
a(n) = Sum_{d|n} d*(binomial(n,d) mod 2). - Ridouane Oudra, Apr 09 2025

cp's OEIS Frontend

A359082 Indices of records in A246600.

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A359080 Numbers k such that A246600(k) = A000005(k).

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A359083 Numbers k such that A246600(k) = A000005(k) and A000005(k) sets a new record.

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A359081 a(n) is the least number k such that A246600(k) = n, and -1 if no such k exists.

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A361937 Numbers k with record values of the ratio A000005(k)/A246600(k) between the total number of divisors of k and the number of divisors d of k such that the bitwise OR of k and d is equal to k.

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A363690 Numbers k such that A246600(k) = 2.

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A363691 Odd numbers k such that A246600(k) = 2.

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

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