A080948 -id:A080948 - cp's OEIS frontend

A080942 Number of divisors of n that are also suffixes of n in binary representation.

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Feb 25 2003

a(n) = 1 iff n = 2^k (A000079), the only divisor is n itself.

For a(n) > 1 the other trivial divisor is 1 for odd numbers and 2 for even numbers (A057716).

			n=63 has A000005(63)=6 divisors: 1='1', 3='11', 7='111', 9='1001', 21='10101' and 63='111111', {1,11,111,111111} are also suffixes of 111111, therefore a(63)=4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007088, A080940, A080941, A080943, A080945, A080946, A080947, A080948.

Cf. A000005, A027750, A030308, A057716, A239826.

import Data.List (isPrefixOf); import Data.Function (on)
a080942 n = length $
            filter ((flip isPrefixOf `on` a030308_row) n) $ a027750_row n
-- Reinhard Zumkeller, Mar 27 2014

a[n_] := DivisorSum[n, 1 &, Mod[n, 2^BitLength[#]] == # &]; Array[a, 100] (* Amiram Eldar, Apr 07 2023 *)

from sympy import divisors
def A080942(n): return sum(1 for d in divisors(n,generator=True) if not (d^n)&((1<Chai Wah Wu, Jun 20 2023

a(A080943(n)) = 2.
a(A080945(n)) > 2.
a(A080946(n)) = 3.
a(A080947(n)) > 3.
a(n) <= A000005(n).
a(p) = 2 for odd primes p.
a(A080948(n)) = n and a(m) < n for m < A080948(n).

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... .

cp's OEIS Frontend

A080942 Number of divisors of n that are also suffixes of n in binary representation.

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