A359083 -id:A359083 - 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)

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, ", "))); }

A363692 Terms of A363690 with a record number of divisors.

Original entry on oeis.org

3, 6, 12, 24, 36, 48, 72, 144, 168, 288, 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

Offset: 1

Amiram Eldar, Jun 16 2023

The corresponding record values are 2, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 32, 40, 48, ... .

David A. Corneth, Table of n, a(n) for n = 1..160 (first 56 terms from Amiram Eldar)

Cf. A000005, A359082, A359083, A361937, A363690, A363693.

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}]; s]; seq[10^5]

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

a(n) <= 2*a(n-1) for n >= 2. - David A. Corneth, Jun 18 2023

A363693 Terms of A363691 with a record number of divisors.

Original entry on oeis.org

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, ", "))); }

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A359082 Indices of records in A246600.

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

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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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A363692 Terms of A363690 with a record number of divisors.

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A363693 Terms of A363691 with a record number of divisors.

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