OFFSET
1,1
COMMENTS
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Binary
Eric Weisstein's World of Mathematics, Prime Partition.
FORMULA
a(n) = Sum_{i=0..L(n)-1} b(i)*prime(i+1) where L=A070939 and b is defined by n = Sum_{i=0..L(n)-1} b(i)*2^i.
G.f.: 1/(1-x) * Sum_{k>=0} prime(k+1)*x^2^k/(1+x^2^k).
log n log log n << a(n) << log^2 n log log n. - Charles R Greathouse IV, Sep 23 2012
For n >= 8, a(n) <= m*(m+1)*(log(m)+log(log(m)))/2 where m = ceiling(log_2(n)). - Robert Israel, Jun 08 2015
EXAMPLE
n=25 -> '11001': a(25) = 1*11 + 1*7 + 0*5 + 0*3 + 1*2 = 20.
This sequence regarded as a triangle with rows of lengths 1, 2, 4, 8, 16, ...:
2
3, 5
5, 7, 8, 10
7, 9, 10, 12, 12, 14, 15, 17
11, 13, 14, 16, 16, 18, 19, 21, 18, 20, 21, 23, 23, 25, 26, 28
13, ... - Philippe Deléham, Jun 07 2015
MAPLE
f:= proc(n) local L, j;
L:= convert(n, base, 2);
add(L[i]*ithprime(i), i=1..nops(L))
end proc:
map(f, [$1..100]); # Robert Israel, Jun 08 2015
MATHEMATICA
a[n_] := With[{bb = IntegerDigits[n, 2]}, bb.Prime[Range[Length[bb], 1, -1]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 27 2021 *)
PROG
(PARI) a(n)=my(v=Vecrev(binary(n)), s, i); forprime(p=2, prime(#v), s+=v[i++]*p); s \\ Charles R Greathouse IV, Sep 23 2012
(Haskell)
a089625 n = f n 0 a000040_list where
f 0 y _ = y
f x y (p:ps) = f x' (y + p * r) ps where (x', r) = divMod x 2
-- Reinhard Zumkeller, Oct 03 2012
(Python)
from sympy import nextprime
def A089625(n):
c, p = 0, 2
while n:
if n&1:
c += p
n >>=1
p = nextprime(p)
return c # Chai Wah Wu, Aug 09 2023
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Reinhard Zumkeller, Dec 31 2003
STATUS
approved