A336510 a(n) = Sum_{p | A055204(n)} 2^(pi(p) - 1).

0, 1, 3, 3, 7, 4, 12, 13, 13, 8, 24, 26, 58, 51, 53, 53, 117, 116, 244, 240, 250, 235, 491, 488, 488, 457, 459, 451, 963, 964, 1988, 1989, 2007, 1942, 1946, 1946, 3994, 3867, 3897, 3900, 7996, 7991, 16183, 16167, 16163, 15906, 32290, 32288, 32288, 32289, 32355

Offset: 1

Author

Michael De Vlieger, Sep 18 2020

Keywords

• nonn
• easy

Comments

All terms of A055204 are squarefree by definition, therefore we can compress the terms of A055204 by interpreting the terms of reverse(A067255(A055204(n))) as a binary number and converted to decimal.

Examples

			A055204(1) = 1, the empty product; by convention a(1) = 0.
5! = 120 = 2^3 * 3 * 5, therefore 2 * 3 * 5 = 30 is the squarefree part, which we write "111", a 1 in the first three places to signify a product of the first three primes. Interpreting "111" as a binary number yields 8. Thus a(5) = 8.
13! = 6227020800 = 2^10 * 3^5 * 5^2 * 7 * 11 * 13; its squarefree part is 3 * 7 * 11 * 13 = 3003, a product of the 2nd, 4th, 5th, and 6th primes. Therefore we write "111010", which, interpreted as a binary number and converted to decimal, is 58. Thus a(13) = 58.
Table illustrating the first terms of this sequence, with b(n) = A055204(n):
               Multiplicities of p|b(n)
   n      b(n)   2  3  5  7 11 13 17 -> Binary   a(n)
  --------------------------------------------------
   1        1    .  .  .  .  .  .  .         0     0
   2        2    1  .  .  .  .  .  .         1     1
   3        6    1  1  .  .  .  .  .        11     3
   4        6    1  1  .  .  .  .  .        11     3
   5       30    1  1  1  .  .  .  .       111     7
   6        5    .  .  1  .  .  .  .       100     4
   7       35    .  .  1  1  .  .  .      1100    12
   8       70    1  .  1  1  .  .  .      1101    13
   9       70    1  .  1  1  .  .  .      1101    13
  10        7    .  .  .  1  .  .  .      1000     8
  11       77    .  .  .  1  1  .  .     11000    24
  12      231    .  1  .  1  1  .  .     11010    26
  13     3003    .  1  .  1  1  1  .    111010    58
  14      858    1  1  .  .  1  1  .    110011    51
  15     1430    1  .  1  .  1  1  .    110101    53
  16     1430    1  .  1  .  1  1  .    110101    53
  17    24310    1  .  1  .  1  1  1   1110101   117
  18    12155    .  .  1  .  1  1  1   1110100   116
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot of the bits of a(n) with (x,y) = (n, a(n)) for 1 <= n <= 2^14.

Crossrefs

Cf. A055204, A067255, A240504.

Programs

Mathematica

Block[{nn = 51, k, p}, k = PrimePi@ nn; Array[Set[p[Prime@ #], 0] &, k]; {0}~Join~Reap[Do[Map[Set[p[#1], Mod[p[#1] + Mod[#2, 2], 2]] & @@ # &, FactorInteger@ i]; Sow[FromDigits[Array[p[Prime[k - # + 1]] &, k], 2]], {i, 2, nn}]][[-1, 1]]] (* or *)
Block[{nn = 51, k = 1}, Reap[Do[Map[If[Mod[k, #] == 0, k /= #, k *= #] &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ i]]; Sow[If[k == 1, 0, Total@ Map[2^(PrimePi[#] - 1) &, FactorInteger[k][[All, 1]] ] ] ], {i, nn}]][[-1, 1]]]