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
Examples
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
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.
Programs
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Mathematica
Table[Total[2^(-1 + Select[Range@ PrimePi[n], OddQ@ Quotient[n, Prime[#]] &])], {n, 50}] -
PARI
a(n) = sum(k=1, primepi(n), if (n\prime(k) % 2, 2^(k-1))); \\ Michel Marcus, Apr 16 2024
Comments