A359081 -id:A359081 - 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... .

A359082 Indices of records in A246600.

Original entry on oeis.org

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)};

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

A359082 Indices of records in A246600.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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