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A093653 Total number of 1's in binary expansion of all divisors of n. 31
1, 2, 3, 3, 3, 6, 4, 4, 5, 6, 4, 9, 4, 8, 9, 5, 3, 10, 4, 9, 9, 8, 5, 12, 6, 8, 9, 12, 5, 18, 6, 6, 8, 6, 9, 15, 4, 8, 10, 12, 4, 18, 5, 12, 15, 10, 6, 15, 7, 12, 9, 12, 5, 18, 11, 16, 10, 10, 6, 27, 6, 12, 17, 7, 8, 16, 4, 9, 10, 18, 5, 20, 4, 8, 16, 12, 11, 20, 6, 15, 12, 8, 5, 27, 9, 10, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 500 terms from Jaroslav Krizek)
Maxwell Schneider and Robert Schneider, Digit sums and generating functions, arXiv:1807.06710 [math.NT], 2018. See (22) p. 6.
FORMULA
a(n) = Sum_{k = 0..n} if(mod(n, k) = 0, A000120(k), 0). - Paul Barry, Jan 14 2005
a(n) = A182627(n) - A226590(n). - Jaroslav Krizek, Sep 01 2013
a(n) = A292257(n) + A000120(n). - Antti Karttunen, Dec 14 2017
From Bernard Schott, May 16 2022: (Start)
If prime p = A000043(n), then a(2^p-1) = a(A000668(n)) = p+1 = A050475(n).
a(2^n) = n+1 (End)
EXAMPLE
a(8) = 4 because the divisors of 8 are [1, 2, 4, 8] and in binary: 1, 10, 100, 1000, so four 1's.
MAPLE
a:= n-> add(add(i, i=Bits[Split](d)), d=numtheory[divisors](n)):
seq(a(n), n=1..100); # Alois P. Heinz, May 17 2022
MATHEMATICA
Table[Plus@@DigitCount[Divisors[n], 2, 1], {n, 75}] (* Alonso del Arte, Sep 01 2013 *)
PROG
(PARI) A093563(n) = sumdiv(n, d, hammingweight(d)); \\ Antti Karttunen, Dec 14 2017
(PARI) a(n) = {my(v = valuation(n, 2), n = (n>>v)); sumdiv(n, d, hammingweight(d)) * (v + 1)} \\ David A. Corneth, Feb 15 2023
(Python)
from sympy import divisors
def a(n): return sum(bin(d).count("1") for d in divisors(n))
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Apr 20 2022
(Python)
from sympy import divisors
def A093653(n): return sum(d.bit_count() for d in divisors(n, generator=True))
print([A093653(n) for n in range(1, 88)]) # Michael S. Branicky, Feb 15 2023
CROSSREFS
Cf. A226590 (number of 0's in binary expansion of all divisors of n).
Cf. A182627 (number of digits in binary expansion of all divisors of n).
Cf. A034690 (a decimal equivalent).
Sequence in context: A087688 A126854 A115206 * A205442 A049982 A245642
KEYWORD
base,easy,nonn
AUTHOR
Jason Earls, May 16 2004
STATUS
approved

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)