A318935 a(n) = Sum_{2^m divides n} 2^(3*m).

1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 4681, 1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 37449, 1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 4681, 1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 299593, 1, 9, 1, 73, 1, 9, 1, 585, 1, 9, 1, 73, 1, 9, 1, 4681

Offset: 1

Author

N. J. A. Sloane, Sep 14 2018

Keywords

• nonn
• easy
• mult

Comments

Sum of cubes of powers of 2 that divide n.

The high-water marks are (8^m - 1)/7, see A023001.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383
• P. J. C. Lamont, The number of Cayley integers of given norm, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See T, p. 102.

Crossrefs

Cf. A007814, A023001.

Programs

Maple

A007814 := n -> padic[ordp](n, 2):
T:= n -> add(2^(3*m),m=0..A007814(n));
[seq(T(n),n=1..100)];

Mathematica

Array[DivisorSum[#, 2^(3 Log2@ #) &, IntegerQ@ Log2@ # &] &, 80] (* or *)
Array[Total[2^(3 Select[Log2@ Divisors@ #, IntegerQ])] &, 80] (* Michael De Vlieger, Nov 07 2018 *)
a[n_] := (8^(IntegerExponent[n, 2] + 1) - 1) / 7; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

PARI

A318935(n) = { my(s=1,w=8); while(!(n%2), s += w; n /= 2; w *= 8); (s); }; \\ Antti Karttunen, Nov 07 2018

Python

from _future_ import division
def A318935(n):
    s = bin(n)
    return (8**(len(s)-len(s.rstrip('0'))+1) - 1)//7 # Chai Wah Wu, Sep 14 2018

Formula

a(n) = (8^(m+1)-1)/7 where m is the 2-adic valuation of n (A007814). - Chai Wah Wu, Sep 14 2018

Thus multiplicative with a(2^m) = (8^(m+1)-1)/7, and a(p^e) = 1 for odd primes p. - Antti Karttunen, Nov 07 2018

Dirichlet g.f.: zeta(s) / (1 - 1/2^(s-3)). - Amiram Eldar, Oct 23 2023

Extensions

Keyword:mult added by Antti Karttunen, Nov 07 2018