A359082 -id:A359082 - cp's OEIS frontend

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

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... .

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, ", "))); }

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, ", "))); }

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

A363692 Terms of A363690 with a record number of divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A363693 Terms of A363691 with a record number of divisors.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI