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%I A371906 #7 Apr 16 2024 13:55:22
%S A371906 0,1,3,2,6,5,13,12,14,11,27,24,56,49,55,54,118,117,245,240,250,235,
%T A371906 491,488,492,461,463,454,966,961,1985,1984,2002,1939,1951,1948,3996,
%U A371906 3869,3903,3898,7994,7985,16177,16160,16166,15911,32295,32292,32300,32297,32363
%N A371906 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is odd.
%C A371906 The only powers of 2 in the sequence are likely 1 and 2.
%H A371906 Michael De Vlieger, <a href="/A371906/b371906.txt">Table of n, a(n) for n = 1..10000</a>
%H A371906 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 A371906 a(1) = 0 since n = 1 is the empty product.
%e A371906 a(2) = 1 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 2^(1-1) = 1.
%e A371906 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 A371906 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 A371906 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 A371906 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 A371906              Power of 2
%e A371906    n   a(n)  01234567      b(n)
%e A371906   ----------------------------
%e A371906    1     0   .               1
%e A371906    2     1   x               2
%e A371906    3     3   xx              6
%e A371906    4     2   .x              3
%e A371906    5     6   .xx            15
%e A371906    6     5   x.x            10
%e A371906    7    13   x.xx           70
%e A371906    8    12   ..xx           35
%e A371906    9    14   .xxx          105
%e A371906   10    11   xx.x           42
%e A371906   11    27   xx.xx         462
%e A371906   12    24   ...xx          77
%e A371906   13    56   ...xxx       1001
%e A371906   14    49   x...xx        286
%e A371906   15    55   xxx.xx       4290
%e A371906   16    54   .xx.xx       2145
%e A371906   17   118   .xx.xxx     36465
%e A371906   18   117   x.x.xxx     24310
%e A371906   19   245   x.x.xxxx   461890
%e A371906   20   240   ....xxxx    46189
%e A371906   ----------------------------
%e A371906                  1111
%e A371906              23571379
%e A371906              Prime factor
%t A371906 Table[Total[2^(-1 + Select[Range@ PrimePi[n], OddQ@ Quotient[n, Prime[#]] &])], {n, 50}]
%o A371906 (PARI) a(n) = sum(k=1, primepi(n), if (n\prime(k) % 2, 2^(k-1))); \\ _Michel Marcus_, Apr 16 2024
%Y A371906 Cf. A008336, A260850, A372000.
%K A371906 nonn,easy
%O A371906 1,3
%A A371906 _Michael De Vlieger_, Apr 15 2024

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