A293431 a(n) is the number of Jacobsthal numbers dividing n.

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

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..21845

Formula

a(n) = Sum_{d|n} A147612(d).

a(n) = A293433(n) + A147612(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A001045(n) = 1.718591611927... . - Amiram Eldar, Jan 01 2024

Example

For n = 15, whose divisors are [1, 3, 5, 15], the first three, 1, 3 and 5 are all in A001045, thus a(15) = 3.
For n = 21, whose divisors are [1, 3, 7, 21], 1, 3 and 21 are in A001045, thus a(21) = 3.
For n = 105, whose divisors are [1, 3, 5, 7, 15, 21, 35, 105], only the divisors 1, 3, 5 and 21 are in A001045, thus a(105) = 4.

Mathematica

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Array[DivisorSum[#, 1 &, MemberQ[s, #] &] &, 105]] (* Michael De Vlieger, Oct 09 2017 *)

Prog

(PARI)
A147612aux(n, i) = if(!(n%2), n, A147612aux((n+i)/2, -i));
A147612(n) = 0^(A147612aux(n, 1)*A147612aux(n, -1));
A293431(n) = sumdiv(n, d, A147612(d));

Crossrefs

Cf. A000005, A001045, A147612, A293432, A293433.

Cf. also A005086.

Sequence in context: A356515 A095955 A272772 * A078573 A143786 A232353

Adjacent sequences: A293428 A293429 A293430 * A293432 A293433 A293434

Keyword

nonn

Author

Antti Karttunen, Oct 09 2017

Status

approved