A293432 Sum of Jacobsthal numbers that divide n.

1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 12, 4, 1, 1, 9, 1, 1, 4, 1, 6, 25, 12, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 15, 1, 6, 4, 1, 1, 4, 6, 1, 25, 44, 12, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 17, 1, 4, 1, 1, 9, 1, 1, 25, 1, 6, 15, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 12, 4, 1, 6, 4, 1, 1, 25, 91, 44, 4, 12, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 15, 6, 1, 4, 1, 1, 30

Offset

1,3

Comments

a(n) is the sum of the divisors of n that are Jacobsthal numbers (A001045).

Links

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

Index entries for sequences related to sums of divisors

Formula

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

a(n) = A293434(n) + (A147612(n)*n).

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) = 1 + 3 + 5 = 9.
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) = 1 + 3 + 5 + 21 = 30.
For n = 21845, whose divisors are [1, 5, 17, 85, 257, 1285, 4369, 21845], the divisors 1, 5, 85 and 21845 are in A001045, thus a(21845) = 1 + 5 + 85 + 21845 = 21936.

Mathematica

With[{s = LinearRecurrence[{1, 2}, {0, 1}, 24]}, Array[DivisorSum[#, # &, 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));
A293432(n) = sumdiv(n, d, A147612(d)*d);
(Python)
from sympy import divisors
def A293432(n): return sum(d for d in divisors(n, generator=True) if (m:=3*d+1).bit_length()>(m-3).bit_length()) # Chai Wah Wu, Apr 18 2025

Crossrefs

Cf. A000203, A001045, A147612, A293431, A293434.

Cf. also A005092.

Sequence in context: A328726 A127556 A227832 * A360327 A050307 A248416

Adjacent sequences: A293429 A293430 A293431 * A293433 A293434 A293435

Keyword

nonn

Author

Antti Karttunen, Oct 09 2017

Status

approved