A359080 -id:A359080 - 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

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, ", "))); }

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))

cp's OEIS Frontend

A359082 Indices of records in A246600.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Python