A371907 a(n) = sum of 2^(k-1) such that floor(n/prime(k)) is even.

0, 0, 0, 1, 1, 2, 2, 3, 1, 4, 4, 7, 7, 14, 8, 9, 9, 10, 10, 15, 5, 20, 20, 23, 19, 50, 48, 57, 57, 62, 62, 63, 45, 108, 96, 99, 99, 226, 192, 197, 197, 206, 206, 223, 217, 472, 472, 475, 467, 470, 404, 437, 437, 438, 418, 427, 297, 808, 808, 815, 815, 1838, 1828

Offset: 1

Author

Michael De Vlieger, Apr 17 2024

Keywords

• nonn
• easy

Comments

This is a transform of A372007(n) = s(n). Write the prime indices of k factors prime(k) | s(n) instead as 2^(k-1) and take the sum for all primes p | s(n). Hence, s(14) = 105 = 3*5*7 becomes a(14) = 2^1 + 2^2 + 2^3 = 2 + 4 + 8 = 14.

Examples

			a(1) = 0 since n = 1 is the empty product.
a(2) = 0 since for n = prime(1) = 2, floor(2/2) = 1 is odd. Therefore a(2) = 0.
a(3) = 0 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) = 0.
a(4) = 1 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(4) = 2^(1-1) = 1.
a(8) = 1 since for n = 8, both floor(8/2) and floor(8/3) are even, but both floor(8/5) and floor(8/7) are odd. Therefore, a(8) = 2^(1-1) + 2^(2-1) = 1 + 2 = 3, etc.
Table relating a(n) with b(n), s(n), and t(n), diagramming powers of 2 with "x" that sum to a(n) or b(n), or prime factors with "x" that produce s(n) or t(n). Sequences s(n) = A372007(n), t(n) = A372000(n), c(n) = A034386(n), b(n) = A371906(n), and c(n) = A357215(n) = a(n) + b(n). Column A (at top) shows powers of 2 that sum to a(n), with B same for b(n), while column S represents prime factors of s(n), T same of t(n).
      [A] 2^k     [B] 2^k
   n   0123  a(n)  012345  b(n)   c(n)   s(n)   t(n)  v(n)
  --------------------------------------------------------
   1   .       0   .         0   2^0-1     1      1   P(0)
   2   .       0   x         1   2^1-1     1      2   P(1)
   3   .       0   xx        3   2^2-1     1      6   P(2)
   4   x       1   .x        2   2^2-1     2      3   P(2)
   5   x       1   .xx       6   2^3-1     2     15   P(3)
   6   .x      2   x.x       5   2^3-1     3     10   P(3)
   7   .x      2   x.xx     13   2^4-1     3     70   P(4)
   8   xx      3   ..xx     12   2^4-1     6     35   P(4)
   9   x       1   .xxx     14   2^4-1     2    105   P(4)
  10   ..x     4   xx.x     11   2^4-1     5     42   P(4)
  11   ..x     4   xx.xx    27   2^5-1     5    462   P(5)
  12   xxx     7   ...xx    24   2^5-1    30     77   P(5)
  13   xxx     7   ...xxx   56   2^6-1    30   1001   P(6)
  14   .xxx   14   x...xx   49   2^6-1   105    286   P(6)
  15   ...x    8   xxx.xx   55   2^6-1     7   4290   P(6)
  16   x..x    9   .xx.xx   54   2^6-1    14   2145   P(6)
  --------------------------------------------------------
       2357        [T] 11
       [S]         235713
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, "Tiger Stripe" Factors of Primorials, ResearchGate, 2024.
• Plot powers 2^(i-1) that sum to a(n) at (x,y) = (n,i) for n = 1..2048, 12X vertical exaggeration.

Crossrefs

Cf. A357215, A371906, A372007.

Programs

Mathematica

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

PARI

a(n) = my(vp=primes([1, n])); vecsum(apply(x->2^(x-1), Vec(select(x->(((n\x) % 2)==0), vp, 1)))); \\ Michel Marcus, Apr 30 2024

Formula

a(n) = A357215(n) - A371906(n).