A371906 - OEIS

%I #7 Apr 16 2024 13:55:22

%S 0,1,3,2,6,5,13,12,14,11,27,24,56,49,55,54,118,117,245,240,250,235,

%T 491,488,492,461,463,454,966,961,1985,1984,2002,1939,1951,1948,3996,

%U 3869,3903,3898,7994,7985,16177,16160,16166,15911,32295,32292,32300,32297,32363

%N a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is odd.

%C The only powers of 2 in the sequence are likely 1 and 2.

%H Michael De Vlieger, <a href="/A371906/b371906.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="https://oeis.org/A372000/a372000.png">Plot powers 2^(i-1) that sum to a(n) at (x,y) = (n,i)</a> for n = 1..2048.

%e a(1) = 0 since n = 1 is the empty product.

%e a(2) = 1 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 2^(1-1) = 1.

%e a(3) = 3 since for n = 3 and prime(1) = 2, floor(3/2) = 1 is odd, and for prime(2) = 3, floor(3/3) = 1 is odd. Hence a(3) = 2^(1-1) + 2^(2-1) = 1 + 2 = 3.

%e a(4) = 2 since for n = 4 and prime(1) = 2, floor(4/2) = 2 is even, but for prime(2) = 3, floor(4/3) = 1 is odd. Therefore, a(n) = 2^(2-1) = 2.

%e a(5) = 6 since for n = 5, though floor(5/2) = 2 is even, floor(5/3) and floor(5/5) are both odd. Therefore, a(n) = 2^(2-1) + 2^(3-1) = 2 + 4 = 6, etc.

%e Table relating a(n) with b(n), diagramming powers of 2 with "x" that sum to a(n), or prime factors with "x" that produce b(n), where b(n) = A372000(n).

%e              Power of 2

%e    n   a(n)  01234567      b(n)

%e   ----------------------------

%e    1     0   .               1

%e    2     1   x               2

%e    3     3   xx              6

%e    4     2   .x              3

%e    5     6   .xx            15

%e    6     5   x.x            10

%e    7    13   x.xx           70

%e    8    12   ..xx           35

%e    9    14   .xxx          105

%e   10    11   xx.x           42

%e   11    27   xx.xx         462

%e   12    24   ...xx          77

%e   13    56   ...xxx       1001

%e   14    49   x...xx        286

%e   15    55   xxx.xx       4290

%e   16    54   .xx.xx       2145

%e   17   118   .xx.xxx     36465

%e   18   117   x.x.xxx     24310

%e   19   245   x.x.xxxx   461890

%e   20   240   ....xxxx    46189

%e   ----------------------------

%e                  1111

%e              23571379

%e              Prime factor

%t Table[Total[2^(-1 + Select[Range@ PrimePi[n], OddQ@ Quotient[n, Prime[#]] &])], {n, 50}]

%o (PARI) a(n) = sum(k=1, primepi(n), if (n\prime(k) % 2, 2^(k-1))); \\ _Michel Marcus_, Apr 16 2024

%Y Cf. A008336, A260850, A372000.

%K nonn,easy

%O 1,3

%A _Michael De Vlieger_, Apr 15 2024