A371906 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is odd.

0, 1, 3, 2, 6, 5, 13, 12, 14, 11, 27, 24, 56, 49, 55, 54, 118, 117, 245, 240, 250, 235, 491, 488, 492, 461, 463, 454, 966, 961, 1985, 1984, 2002, 1939, 1951, 1948, 3996, 3869, 3903, 3898, 7994, 7985, 16177, 16160, 16166, 15911, 32295, 32292, 32300, 32297, 32363

Offset

1,3

Comments

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Plot powers 2^(i-1) that sum to a(n) at (x,y) = (n,i) for n = 1..2048.

Example

a(1) = 0 since n = 1 is the empty product.
a(2) = 1 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 2^(1-1) = 1.
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.
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.
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.
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).
             Power of 2
   n   a(n)  01234567      b(n)
  ----------------------------
   1     0   .               1
   2     1   x               2
   3     3   xx              6
   4     2   .x              3
   5     6   .xx            15
   6     5   x.x            10
   7    13   x.xx           70
   8    12   ..xx           35
   9    14   .xxx          105
  10    11   xx.x           42
  11    27   xx.xx         462
  12    24   ...xx          77
  13    56   ...xxx       1001
  14    49   x...xx        286
  15    55   xxx.xx       4290
  16    54   .xx.xx       2145
  17   118   .xx.xxx     36465
  18   117   x.x.xxx     24310
  19   245   x.x.xxxx   461890
  20   240   ....xxxx    46189
  ----------------------------
                 1111
             23571379
             Prime factor

Mathematica

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

Prog

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

Crossrefs

Cf. A008336, A260850, A372000.

Sequence in context: A301501 A072787 A338524 * A293549 A370377 A306443

Adjacent sequences: A371903 A371904 A371905 * A371907 A371908 A371909

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 15 2024

Status

approved